Slide 27 / 252 Similar Triangle Measuring Device By looking along the meter stick, you can then move the card so that a distant object fills either the 0.5 cm, 2 cm or 4 cm slot. You can then measure how far the card is from your eye, along the meter stick. This creates a similar triangle that allows you to find how far away an object of known size is, or the size of an object of known distance away. Slide 28 / 252 Similar Triangle Measuring Device This shows how by lining up a distant object to fill a slot on the device two similar triangles are created, the small red one and the larger blue one. All the angles are equal and the sides are in proportion. Also, the base and altitude of each isosceles triangle will be in proportion. Slide 29 / 252 Similar Triangle Measuring Device The altitude and base of the small isosceles triangle can be directly measured, which means that the ratio of those on the larger triangle is know. Given the size or the distance to the object, the other can be determined.
Slide 30 / 252 Similar Triangle Measuring Device You are visiting Paris and have your similar triangle measuring device with you. You know that the Eiffel Tower is 324 meters tall. You adjust your device so that turned sidewise the height of the tower fills the 2 cm slot when the card is 20 cm from your eye. How far are you from the tower? Slide 30 (Answer) / 252 Similar Triangle Measuring Device You are visiting Paris and have your similar triangle measuring device with you. Math Practice This example (this slide and You know that the Eiffel Tower is 324 meters tall. the next) addresses MP1, You adjust your device so that turned sidewise the height of the MP2, MP4, MP5 & MP7 tower fills the 2 cm slot when the card is 20 cm from your eye. How far are you from the tower? [This object is a pull tab] Slide 31 / 252 Shadows and Similar Right Triangles distance to tower distance to card = height of tower width of slot distance to card distance to tower = x height of tower width of slot 20 cm d = 2 cm (324 m) d = 3240 m d 20 cm 324 m 2 cm 2 m
Slide 32 / 252 5 You move to another location and the Eiffel Tower (324 m tall) now fills the 4 cm slot when the card is 48 cm from your eye. How far are you from the Eiffel Tower now? Slide 32 (Answer) / 252 5 You move to another location and the Eiffel Tower (324 m tall) now fills the 4 cm slot when the card is 48 cm from your eye. How far are you from the Eiffel Tower now? Answer 3888 m [This object is a pull tab] Slide 33 / 252 6 The tallest building in the world, the Burj Kalifah in Dubai, is 830 m tall. You turn your device so that it fills the 4 cm slot when it is 29.4 cm from your eye. How far are you from the building?
Slide 33 (Answer) / 252 6 The tallest building in the world, the Burj Kalifah in Dubai, is 830 m tall. You turn your device so that it fills the 4 cm slot when it is 29.4 cm from your eye. How far are you from the building? Answer 8200 m [This object is a pull tab] Slide 34 / 252 7 The width of a storage tank fills the 2 cm slot when the card is 48 cm from your eye. You know that the tank is 680 m away. What is its width? Slide 34 (Answer) / 252 7 The width of a storage tank fills the 2 cm slot when the card is 48 cm from your eye. You know that the tank is 680 m away. What is its width? Answer 28 m [This object is a pull tab]
Slide 35 / 252 8 The moon has a diameter of 3480 km. You measure it one night to about fill the 0.5 cm slot when the card is 54 cm from your eye. What is the distance to the moon? Slide 35 (Answer) / 252 8 The moon has a diameter of 3480 km. You measure it one night to about fill the 0.5 cm slot when the card is 54 cm from your eye. What is the distance to the moon? Answer 375,840 km [This object is a pull tab] Slide 36 / 252 Similar Triangles and Trigonometry sinθ 1 θ cosθ Return to the Table of Contents
Slide 37 / 252 Problem Solving Recall that Thales found the height of the pyramid by using similar triangles created by the shadow of the pyramid and a rod of known length. Slide 38 / 252 Problem Solving But what if he were trying to solve this problem and there wasn't a shadow to use. Or you are trying to solve other types of problems which don't allow you to set up a similar triangle so easily. Trigonometry provides the needed similar triangle for any circumstance, which is why it is a powerful tool. Slide 39 / 252 Problem Solving with Trigonometry So, if Thales used trig to solve his problem, he'd have considered this right triangle. First he'd measure theta, the angle between the ground and the top of the pyramid, when at a certain distance away on the ground. Then he'd imagine a similar triangle with the same angle. height θ distance
Slide 40 / 252 Problem Solving with Trigonometry He has a ready-made right triangle, thanks to mathematicians who calculated all the possible right triangles that could be created with a hypotenuse of 1 and put their measurements in a table, a trigonometry table. The side opposite the angle is named sine θ, or sinθ for short, and the side adjacent to the angle is called cosine θ, or cosθ for short. height 1 sinθ θ θ distance cosθ Slide 41 / 252 Problem Solving with Trigonometry We know all the angles are equal since both triangles include a right angle and the angle theta, so those two angles are the same in both. And, since all the angles of a triangle total to 180 º, all three angles must be equal. Since all the angles are equal, these triangles are similar. height 1 sinθ θ θ distance cosθ Slide 42 / 252 Problem Solving with Trigonometry Since all the angles are equal, the sides are in proportion, so what would this ratio be equal to in the triangle to the right? height sin θ = distance cos θ height 1 sin θ θ θ distance cos θ
Slide 42 (Answer) / 252 Problem Solving with Trigonometry Since all the angles are equal, the sides are in proportion, so what would this ratio be equal to in the triangle to the right? Math Practice The question on this slide addresses MP2. height sin θ = distance cos θ [This object is a pull tab] height 1 sin θ θ θ distance cos θ Slide 43 / 252 Problem Solving with Trigonometry When we did the problem earlier we used the rod's height of 1 m and it's shadow's length of 2 m. That would mean that the angle between the rays of sunlight and the ground would have been 26.6º. And the length of the pyramid's shadow was 120 m. Let's use that angle and distance and see if we get the same answer. height 1 sinθ 26.6º 26.6º 120 m cosθ Slide 44 / 252 Problem Solving with Trigonometry If the distance was 120 m, and the angle was 26.6º, you find the height by solving for it and then using your calculator to look up the values for sin and cos. height sinθ = distance cosθ height sin(26.6º) = 120 m cos(26.6º) sin(26.6º) height = (120 m) cos(26.6º) (0.448) = (120 m) = 60 m (0.894) height 1 sinθ 26.6º 26.6º 120 m cosθ
Slide 45 / 252 Tangent θ height sinθ Early in the last problem we found that: = distance cosθ This ratio of sine to cosine is used very often, and has its own name: Tangent θ, or tanθ for short. Tangent θ is defined as Sine θ divided by Cosine θ. sinθ tanθ = cosθ Slide 46 / 252 Using Calculators with Trigonometry The last step of that problem required finding the values of the sine and the cosine of 26.6º. When working with trigonometry, you'll need to find the values of sine, cosine and other trig functions when given an angle. This used to involve using tables, but now it's pretty simple to use a basic scientific calculator. Slide 47 / 252 Using Calculators with Trigonometry Basic scientific calculators are available on computers, tablets and smart phones. They can also be a separate device, similar to the inexpensive calculator shown here. It can do everything you'll need for this course.
Slide 48 / 252 Using Calculators with Trigonometry The trig functions we're going to be using right now are sine, cosine and tangent. Those are marked in the box on the calculator. On most calculators, they are noted by buttons which say SIN COS TAN Slide 49 / 252 Using Calculators with Trigonometry This is for finding the sine of an angle. Slide 50 / 252 Using Calculators with Trigonometry This is for finding the cosine of an angle.
Slide 51 / 252 Using Calculators with Trigonometry This is for finding the tangent of an angle. Slide 52 / 252 Problem Solving with Trigonometry 1 sinθ θ cosθ Slide 53 / 252 Inclinometer In practice, we often have to measure angles of elevation or depression in order to solve problems. There are very accurate ways of doing that which are used by surveyors, navigators and others. But you can make a simple device, called an inclinometer, to accomplish the same thing, and then solve problems on your own.
Slide 54 / 252 Inclinometer Just tape a protractor to a meter stick and hang a small weight from the hole in the protractor. Set it up so that when the meter stick is horizontal, the string goes straight down. Slide 55 / 252 Inclinometer Then, if you look along the meter stick, you can hold the string where it touches the protractor and read the angle. You'll have to subtract 90 degrees to get the angle to the horizon, or angle of elevation. Slide 56 / 252 Inclinometer You are standing on the ground and look along your inclinometer to see the top of a building to be at an angle of 30º. You then measure the distance to the base of the building to be 30 m. Find the height of the building, remembering to add in the height your eye is above the ground.
Slide 56 (Answer) / 252 Inclinometer You are standing on the ground and look along your height inclinometer to see the top of a building to be at an angle distance = tanθ of 30º. You then measure the distance to the base of the building to be 30 m. Find the height of the building, height = tanθ x distance remembering to add in the height your eye is above the h = tan(30º)(30m) ground. Answer h = (0.577)(30m) h = 17.32 m + 1.52m h = 18.84 m Note: if your eyesight is 5 ft above the ground, that is 1.52 m [This object is a pull tab] Feet to meters = 5/3.28 = 1.52 Slide 57 / 252 Example You are standing 200 m away from the base of a building. You measure the top of the building to be at an angle of elevation (the angle between the ground and a line drawn to the top) of 60º. What is the height of the height building? 60 º 200 m Slide 57 (Answer) / 252 Example You are standing 200 m away from the base of a building. You measure the top of the building to be at an angle of Math Practice This example (this slide and elevation (the angle between the the next 2) addresses MP4 & ground and a line drawn to the top) of 60º. MP5. What is the height of the height building? [This object is a pull tab] 60 º 200 m
Slide 58 / 252 Example Make a quick sketch showing the original right triangle and one showing the appropriate trig functions. height 1 sin(60º) 60º 60º 200 m cos(60º) Slide 59 / 252 Example Then set up the ratios, substitute the values and solve. height 1 sin(60 º) 60 º 60 º 200 m cos(60 º) Slide 59 (Answer) / 252 Example Then set up the ratios, substitute the values and solve. height = tanθ distance height = tanθ x distance Answer h = tan(60º)(200m) h = (1.73)(200m) height h = 346 m [This object is a pull tab] 1 sin(60 º) 60 º 60 º 200 m cos(60 º)
Slide 60 / 252 9 You are standing 30 m away from the base of a building. The top of the building lies at an angle of elevation (the angle between the ground and the hypotenuse) of 50º. What is the height of the building? height 50º 30 m Slide 60 (Answer) / 252 9 You are standing 30 m away from the base of a building. The top of the building lies at an angle of elevation (the angle between the ground and the hypotenuse) of 50º. What is the height of the building? Answer 36 m height [This object is a pull tab] 50º 30 m Slide 61 / 252 10 You are standing 50 m away from the base of a building. The building creates an angle of elevation with the ground measuring 80º. What is the height of the building?
Slide 61 (Answer) / 252 10 You are standing 50 m away from the base of a building. The building creates an angle of elevation with the ground measuring 80º. What is the height of the building? Answer 283 m [This object is a pull tab] Slide 62 / 252 11 Use the tanθ function of your calculator to determine the height of a flagpole if it is 30 m away and it's angle of elevation with the ground measures 70º. Slide 62 (Answer) / 252 11 Use the tanθ function of your calculator to determine the height of a flagpole if it is 30 m away and it's angle of elevation with the ground measures 70º. Answer 82 m [This object is a pull tab]
Slide 63 / 252 12 Use the tanθ function of your calculator to determine the height of a building if its base is 50 m away and it's angle of elevation with the ground measures 20º. Slide 63 (Answer) / 252 12 Use the tanθ function of your calculator to determine the height of a building if its base is 50 m away and it's angle of elevation with the ground measures 20º. Answer 18 m [This object is a pull tab] Slide 64 / 252 13 You are on top of a building and look down to see someone who standing the ground. The angle of depression (the angle below the horizontal to an object) is 30º and they are 90 m from the base of the building. How high is the building? (Neglect the heights of you and the other person.) Make sure to draw a sketch!
Slide 64 (Answer) / 252 13 You are on top of a building and look down to see someone who standing the ground. The angle of depression (the angle below the horizontal to an object) is 30º and they are 90 m from the base of the building. How high is the building? (Neglect the heights of you and the other person.) Answer Make sure to draw a sketch! 52 m [This object is a pull tab] Slide 65 / 252 14 Determine the distance an object lies from the base of a 45 m tall building if the angle of depression to it is 40º. Slide 65 (Answer) / 252 14 Determine the distance an object lies from the base of a 45 m tall building if the angle of depression to it is 40º. Answer 54 m [This object is a pull tab]
Slide 66 / 252 Trigonometric Ratios When solving problems with trig, you find a right triangle which is similar to the one below. Then you find the solution by setting up the ratios of proportion. But, since the hypotenuse is 1, often it's forgotten that these are ratios. sinθ 1 θ cosθ Slide 67 / 252 Trigonometric Ratios Return to the Table of Contents Slide 68 / 252 Trigonometric Ratios Fill in the fundamental trig ratios below: Sine called "sin" for short click Cosine called "cos" for short click Tangent called "tan" for short click
Slide 69 / 252 Trigonometric Ratios The name of the angle usually follows the trig function. If the angle is named θ (theta) the names become: · sinθ · cosθ · tanθ If the angle is named α (alpha) the functions become: · sinα · cosα · tanα Slide 70 / 252 Trigonometric Ratios If you have the sides, trig ratios let you find the angles. But if you have a side and an angle, trig ratios also let you find the other sides. Slide 71 / 252 Trigonometric Ratios These ratios depend on which angle you are calling θ; never hypotenuse the right angle. opposite You know that the side side opposite the right angle is called the hypotenuse. θ The leg opposite θ is called adjacent side the opposite side. The leg that touches θ is called the adjacent side. sinθ 1 θ cosθ
Slide 72 / 252 Trigonometric Ratios There are two possible angles that can be called # . θ Once you choose which angle hypotenuse is # , the names of the sides are defined. adjacent side You can change later, but then the names of the sides also change. opposite side Slide 73 / 252 Trigonometric Ratios With this theta, these become the sides. sinθ 1 θ cosθ Slide 74 / 252 Trigonometric Ratios α If you use the other angle, named α here, the names change accordingly. 1 cosα sinα
Slide 75 / 252 Trigonometric Ratios Let's say I'm solving a problem that involves this right triangle. To use trig, I'd find a right triangle with hypotenuse of 1 hypotenuse and legs of sinθ and cosθ opposite which has the same angle θ side so, it's similar. θ adjacent side Slide 76 / 252 Trigonometric Ratios Then set up the ratios. There are basic ratios relating hypotenuse the sides of these two opposite triangles. side Since they are similar θ triangles, the ratio of any two sides in one triangle is equal adjacent side to that ratio of sides in the other. sinθ 1 θ cosθ Slide 77 / 252 Trigonometric Ratios hypotenuse opposite sinθ opposite side opp side = = 1 hypotenuse hyp θ adjacent side sinθ 1 θ cosθ
Slide 78 / 252 Trigonometric Ratios hypotenuse opposite side θ cosθ adjacent side adj = = adjacent side 1 hypotenuse hyp sinθ 1 θ cosθ Slide 79 / 252 Trigonometric Ratios hypotenuse opposite side θ adjacent side sinθ opposite side opp = = cosθ adjacent side adj sinθ 1 θ cosθ Slide 80 / 252 Trigonometric Ratios hypotenuse opposite sinθ opposite side opp side = = 1 hypotenuse hyp θ cosθ adjacent side adj = = adjacent side 1 hypotenuse hyp sinθ opposite side opp = = cosθ adjacent side adj sinθ 1 θ cosθ
Slide 81 / 252 Trigonometric Ratios But these can be simplified since: sinθ opposite side opp sinθ = = = sinθ 1 hypotenuse hyp 1 cosθ cosθ adjacent side adj = cosθ = = 1 1 hypotenuse hyp sinθ sinθ opposite side opp cosθ = tanθ = = cosθ adjacent side adj Slide 82 / 252 Trigonometric Ratios hypotenuse opposite opposite side opp side sinθ = = hypotenuse hyp θ adj adjacent side cosθ = = hyp adjacent side hypotenuse opposite side opp = = tanθ adj adjacent side sinθ 1 θ cosθ Slide 83 / 252 Trigonometric Ratios These trig ratios are used so often that they are memorized with the expression "SOH CAH TOA." opposite side o pp = = s inθ SOH hypotenuse h yp adjacent side a dj CAH c osθ = = h yp hypotenuse opposite side o pp TOA t anθ = = adjacent side a dj If you get confused w/ the vowel sounds in SOH CAH TOA, you could also try the mnemonic sentence below. S ome O ld H orse C aught A nother H orse T aking O ats A way.
Slide 83 (Answer) / 252 Trigonometric Ratios These trig ratios are used so often that they are memorized with the expression "SOH CAH TOA." MP6: Throughout this lesson, emphasize the correct set up of the opposite side o pp trig ratios. s inθ = = SOH hypotenuse h yp Math Practice Q's to help w/ this include: adjacent side a dj What side lengths am I given? CAH c osθ = = h yp hypotenuse (MP1) How are the sides related to the o pp opposite side angle? (MP7) TOA = = t anθ adjacent side a dj Which trig ratio am I going to use? (MP6) Construct the equation to solve this If you get confused w/ the vowel sounds in SOH CAH TOA, you problem. (MP2) could also try the mnemonic sentence below. [This object is a pull tab] S ome O ld H orse C aught A nother H orse T aking O ats A way. Slide 84 / 252 15 Find the sin θ . Round your answer to the nearest hundredth. θ 8.5 3.0 8.0 Slide 84 (Answer) / 252 15 Find the sin θ . Round your answer to the nearest hundredth. θ Answer 8.5 3.0 0.94 8.0 [This object is a pull tab]
Slide 85 / 252 16 Find the cos θ . Round your answer to the nearest hundredth. θ 8.5 3.0 8.0 Slide 85 (Answer) / 252 16 Find the cos θ . Round your answer to the nearest hundredth. θ Answer 8.5 0.35 3.0 [This object is a pull tab] 8.0 Slide 86 / 252 17 Find the tan θ . Round your answer to the nearest hundredth. θ 8.5 3.0 8.0
Slide 86 (Answer) / 252 17 Find the tan θ . Round your answer to the nearest hundredth. θ Answer 2.67 8.5 3.0 [This object is a pull tab] 8.0 Slide 87 / 252 18 Find the tan θ . Round your answer to the nearest hundredth. θ 16 7 14 Slide 87 (Answer) / 252 18 Find the tan θ . Round your answer to the nearest hundredth. θ Answer 16 2 7 [This object is a pull tab] 14
Slide 88 / 252 19 Find the sin θ . Round your answer to the nearest hundredth. θ 16 7 14 Slide 88 (Answer) / 252 19 Find the sin θ . Round your answer to the nearest hundredth. θ Answer 16 0.88 7 [This object is a pull tab] 14 Slide 89 / 252 20 Find the cos θ . Round your answer to the nearest hundredth. θ 16 7 14
Slide 89 (Answer) / 252 20 Find the cos θ . Round your answer to the nearest hundredth. θ Answer 16 0.44 7 [This object is a pull tab] 14 Slide 90 / 252 Trigonometric Ratios For instance, let's find the length of side x. 7.0 The side we're looking for is x opposite the given angle; and the given length is the hypotenuse; 30º so we'll use the trig function that relates these three: opp opposite side = sinθ = hyp hypotenuse Slide 91 / 252 Trigonometric Ratios opposite side opp sinθ = = hypotenuse hyp sinθ = opp hyp 7.0 x opp = (hyp) (sinθ) x = (7.0)(sin(30º)) 30º x = (7.0)(0.50) x = 3.5
Slide 92 / 252 Trigonometric Ratios Now, let's find the length of side x in this case. The side we're looking for is 9.0 adjacent the given angle; and the given length is the hypotenuse; 25º so we'll use the trig function x that relates these three: adjacent side adj cosθ = = hypotenuse hyp Slide 93 / 252 Trigonometric Ratios adjacent side adj cosθ = = hypotenuse hyp adj cosθ = hyp 9.0 adj = (hyp)(cosθ) x = (9.0)(cos(25º)) 25º x = (9.0)(0.91) x x = 8.2 Slide 94 / 252 Trigonometric Ratios Now, let's find the length of side x in this case. 50º The side we're looking for is 9.0 adjacent the given angle; and the given length is the opposite the given angle; so we'll use the trig function x that relates these three: adjacent side = opp opposite side tanθ = adj
Slide 95 / 252 Trigonometric Ratios tanθ = opposite side adjacent side= opp adj 50º opp tanθ = adj 9.0 opp = (adj)(tanθ) x = (9.0)(tan(50º)) x = (9.0)(1.2) x x = 10.8 Slide 96 / 252 21 Find the value of x. Round your answer to the nearest tenth. 35 64º x Slide 96 (Answer) / 252 21 Find the value of x. Round your answer to the nearest tenth. Answer 35 17.1 [This object is a pull tab] 64º x
Slide 97 / 252 22 Find the value of x. Round your answer to the nearest tenth. x 28 36º Slide 97 (Answer) / 252 22 Find the value of x. Round your answer to the nearest tenth. Answer 47.6 x 28 [This object is a pull tab] 36º Slide 98 / 252 23 Find the value of x. Round your answer to the nearest tenth. 44º 28 x
Slide 98 (Answer) / 252 23 Find the value of x. Round your answer to the nearest tenth. Answer 19.5 44º [This object is a pull tab] 28 x Slide 99 / 252 24 Find the value of x. Round your answer to the nearest tenth. 7.4 37º x Slide 99 (Answer) / 252 24 Find the value of x. Round your answer to the nearest tenth. Answer 5.9 7.4 37º [This object is a pull tab] x
Slide 100 / 252 Applications of Trigonometric Ratios Most of the time, trigonometric ratios are used to solve real- world problems, as you saw at the beginning of this unit. Now that you are familiar with the derivation of the three trigonometric ratios (sine, cosine, and tangent), you are ready to apply your knowledge and practice solving these problems. Before we begin, let's review some key vocabulary that you will see in these word problems. Slide 100 (Answer) / 252 Applications of Trigonometric Ratios Most of the time, trigonometric ratios are used to solve real- These next 12 slides address world problems, as you saw at the beginning of this unit. MP1, MP2, MP4, MP5, MP6 & Now that you are familiar with the derivation of the three Math Practice MP7. trigonometric ratios (sine, cosine, and tangent), you are ready to apply your knowledge and practice solving these Use the questions provided in problems. the "Math Practice" pull tab on Before we begin, let's review some key vocabulary that you slide #98 to provide additional will see in these word problems. assistance, when needed. [This object is a pull tab] Slide 101 / 252 Applications of Trigonometric Ratios The angle of elevation is the The angle of depression is the angle above the horizontal to angle below the horizontal to an object. an object. object observer sight line angle of depression sight line angle of elevation object
Slide 102 / 252 Applications of Trigonometric Ratios The angle of elevation and the angle of depression are both measured relative to parallel horizontal lines, so they are equal in measure. f 20 º o n e o l g i s n s a e r p e 10,000ft d angle of elevation 20 º Slide 103 / 252 Applications of Trigonometric Ratios Example Amy is flying a kite at an angle of 58º. 158 feet The kite's string is 1 58 feet long and x Amy's arm is 3 feet off the ground. 58 o How high is the kite off the ground? 3 feet Slide 104 / 252 Applications of Trigonometric Ratios x sinθ = 158 x sin58 = 158 158ft x x .8480 = 158 x = 134 58º Now, we must add in Amy's arm height. 134 + 3 = 137 The kite is about 137 feet off the ground.
Slide 105 / 252 Applications of Trigonometric Ratios Example You are standing on a mountain that is 5306 feet high. You look down at your campsite at angle of 30 º. If you are 6 feet tall, how far is the base of the mountain from the campsite? 30 o 6 ft 5306 ft x Slide 106 / 252 Applications of Trigonometric Ratios 5312 tan30 = x 5312 .5774 = 5312 ft x 30º .5774x = 5312 x x ≈ 9,200 ft The campsite is about 9,200 ft from the base of the mountain. Slide 107 / 252 Applications of Trigonometric Ratios Example: Vernon is on the top deck of a cruise ship and observes 2 dolphins following each other directly away from the ship in a straight line. Veron's position is 154 m above sea level, and the angles of depression to the 2 dolphins to the ship are 35 º and 36º, respectively. Find the distance between the 2 dolphins to the nearest hundredth of a meter. 154 m
Slide 108 / 252 Applications of Trigonometric Ratios The first step is to divide the diagram into two separate ones. Then, find the horizontal distance in both. Let's call them x & y. 154 m 35º x 154 m 36º y Then, use your trigonometric ratios to find these values. Slide 109 / 252 Applications of Trigonometric Ratios 154 m 35º x tan 35 = 154 x 154 0.7002 = x 0.7002x = 154 x = 219.94 m Slide 110 / 252 Applications of Trigonometric Ratios 154 m 36º y tan 36 = 154 y 154 0.7265 = y 0.7265y = 154 y = 211.98 m
Slide 111 / 252 Applications of Trigonometric Ratios 154 m 219.94 m 211.98 m Now, if we subtract these measurements, then we will find the distance between the 2 dolphins. 219.94 - 211.98 = 7.96 m Slide 112 / 252 25 You are looking at the top of a tree. The angle of elevation is 55º. The distance from the top of the tree to your position (line of sight) is 84 feet. If you are 5.5 feet tall, how far are you from the base of the tree? Slide 112 (Answer) / 252 25 You are looking at the top of a tree. The angle of elevation is 55º. The distance from the top of the tree to your position (line of sight) is 84 feet. If you are 5.5 feet 84 ft tall, how far are you from the base of the tree? 55 o Answer x x cos55 = 84 x = 48.18 You are approximately 48 ft from the base of the tree. [This object is a pull tab]
Slide 113 / 252 26 A wheelchair ramp is 3 meters long and inclines at 6º. Find the height of the ramp to the nearest hundredth of a centimeter. Slide 113 (Answer) / 252 26 A wheelchair ramp is 3 meters long and inclines at 6º. Find the height of the ramp to the nearest hundredth of a centimeter. 6 o Answer y sin6 = 3 y = 0.3136 m The height of the ramp is 31.36 cm [This object is a pull tab] Slide 114 / 252 27 John wants to find the height of a building which is casting a shadow of 175 ft at an angle of 73.75º. Find the height of the building to the nearest foot.
Slide 114 (Answer) / 252 27 John wants to find the height of a building which is casting a shadow of 175 ft at an angle of 73.75º. Find the height of the building to the nearest foot. h 73.75 o Answer 175 ft h tan 73.75 = 175 h = 600 The height of the building is 600 ft [This object is a pull tab] Slide 115 / 252 28 A sonar operator on a ship detects a submarine that is located 800 meters away from the ship at an angle of depression of 38º. How deep is the submarine? 38° 800 m Slide 115 (Answer) / 252 28 A sonar operator on a ship detects a submarine that is located 800 meters away from the ship at an angle of depression of 38º. How deep is the submarine? Answer 492.53 meters 38° 800 m [This object is a pull tab]
Slide 116 / 252 29 A sonar operator on a ship detects a submarine that is located 800 meters away from the ship at an angle of depression of 38º. If the submarine stays in the same position, then how far would the ship need to travel to be directly above the submarine? 38° 800 m Slide 116 (Answer) / 252 29 A sonar operator on a ship detects a submarine that is located 800 meters away from the ship at an angle of depression of 38º. If the submarine stays in the same position, then how far would the ship need to travel to be directly above the submarine? Answer 38° 630.41 meters 800 m [This object is a pull tab] Slide 117 / 252 30 The ship is traveling at a speed of 32 meters per second, in the direction towards the submarine. From its current position, how many minutes, to the nearest tenth of a minute, will it take the ship to be directly over the submarine. 38° 800 m
Slide 117 (Answer) / 252 30 The ship is traveling at a speed of 32 meters per second, in the direction towards the submarine. From its current position, how many minutes, to the nearest tenth of a minute, will it take the ship to be directly over the submarine. distance to get above the submarine Answer is 630.41 meters (found in last 38° question) 800 m 630.41/32 = 19.70 seconds 19.70/60 = 0.3 minutes [This object is a pull tab] Slide 118 / 252 Inverse Trigonometric Ratios Return to the Table of Contents Slide 119 / 252 Inverse Trigonometric Ratios So far, you have used the sine, cosine, and tangent ratios when given the measurement of the acute angle θ in a right triangle to find the measurements of the missing sides. What can you use when you need to find the measurements of the acute angles? We have what are called the inverse sine, inverse cosine and inverse tangent ratios that will help us answer the question above. If you know the measures of 2 sides of a triangle, then you can find the measurement of the angle with these ratios.
Slide 120 / 252 Inverse Trigonometric Ratios Remember: The Inverse Trigonometric Ratios are given below . o o a S C T h h a ( ) A opp opp If sinθ = , θ = sin -1 hyp hyp ( ) adj adj hyp If cosθ = , θ = cos -1 opp hyp hyp opp ( ) opp θ If tanθ = , θ = tan -1 adj B adj adj C Slide 120 (Answer) / 252 Inverse Trigonometric Ratios Remember: The Inverse Trigonometric Ratios are given below . o a o MP6: Throughout this lesson, S C T h a A h ( ) opp emphasize the correct set up of the opp If sinθ = , θ = sin -1 hyp hyp trig ratios. ( ) adj adj Q's to help w/ this include: hyp Math Practice If cosθ = , θ = cos -1 opp hyp hyp What side lengths am I given? (MP1) opp ( ) How are the sides related to the opp θ If tanθ = , θ = tan -1 adj B adj angle? (MP7) C adj Which trig ratio am I going to use? (MP6) Construct the equation to solve this problem. (MP2) [This object is a pull tab] Slide 121 / 252 Using Calculators with Inverse Trigonometry The inverse trig functions are located just above the sine, cosine and tangent buttons. They are marked in the box on the calculator. On most calculators, they are noted by text which says SIN -1 COS -1 TAN -1 In most cases, they can be used by pressing the 2nd, or shift, button (arrow pointing to it) & the sine, cosine, or tangent button.
Slide 122 / 252 31 Find sin -1 (0.8) Round the angle measure to the nearest hundredth. Slide 122 (Answer) / 252 31 Find sin -1 (0.8) Round the angle measure to the nearest hundredth. Answer θ = 53.13° [This object is a pull tab] Slide 123 / 252 32 Find tan -1 (2.3). Round the angle measure to the nearest hundredth.
Slide 123 (Answer) / 252 32 Find tan -1 (2.3). Round the angle measure to the nearest hundredth. Answer θ = 66.50° [This object is a pull tab] Slide 124 / 252 33 Find cos -1 (0.45). Round the angle measurement to the nearest hundredth. Slide 124 (Answer) / 252 33 Find cos -1 (0.45). Round the angle measurement to the nearest hundredth. Answer θ = 63.26° [This object is a pull tab]
Slide 125 / 252 Inverse Trigonometric Ratios To find an unknown angle measure in a right triangle, you need to identify the correct trig function that will find the missing value. Use "SOH CAH TOA" to help. 9 ∠ A is your angle of reference. A B Label the two given sides θ of your triangle opp, adj, or hyp. Identify the trig funtion that uses ∠ A, 15 and the two sides. C Slide 126 / 252 Inverse Trigonometric Ratios Using "SOH CAH TOA", I have "a" and "h", so the ratio is a/h which is cosine. 9 adj A B θ hyp 15 C now you can solve for m ∠ A, the missing angle cos A = 9 15 using the inverse trig function. ( ) m ∠ A = cos -1 9 15 Once you find m ∠ A, you can easily find m ∠ C, m ∠ A = 53.13º using the Triangle Sum Theorem. Slide 127 / 252 Inverse Trigonometric Ratios Now, let's find the measurement of the angle θ in θ this case. 13 The sides that we are given are the opposite side & the hypotenuse; 12 so we'll use the trig function that relates these two sides with our angle: opposite side opp sinθ = = hyp hypotenuse
Slide 128 / 252 Inverse Trigonometric Ratios θ sin θ = 12 13 13 ( ) θ = sin -1 12 13 12 θ = 67.38º Slide 129 / 252 34 Find the m ∠ D in the figure below. D 13 F E 23 Slide 129 (Answer) / 252 34 Find the m ∠ D in the figure below. D Answer θ = 60.52° 13 F E 23 [This object is a pull tab]
Slide 130 / 252 35 Find the m ∠ F in the figure below. D 35 F E 27 Slide 130 (Answer) / 252 35 Find the m ∠ F in the figure below. D Answer θ = 39.52° 35 F E 27 [This object is a pull tab] Slide 131 / 252 36 Find the m ∠ G in the figure below. G 18 J H 17
Slide 131 (Answer) / 252 36 Find the m ∠ G in the figure below. G 18 Answer θ = 70.81° J H 17 [This object is a pull tab] Slide 132 / 252 Applications of Inverse Trigonometric Ratios As we discussed earlier in this unit, trigonometric ratios and the inverse trigonometric ratios are used to solve real-world problems. Now that you are familiar with the three inverse trigonometric ratios (inverse sine, inverse cosine, and inverse tangent), you are ready to apply your knowledge and practice solving these problems. Slide 132 (Answer) / 252 Applications of Inverse Trigonometric Ratios These next few examples (the next 4 slides) address MP1, As we discussed earlier in this unit, trigonometric ratios and MP2, MP4, MP5, MP6 & MP7. the inverse trigonometric ratios are used to solve real-world Math Practice problems. Use the questions provided in Now that you are familiar with the three inverse trigonometric the "Math Practice" pull tab on ratios (inverse sine, inverse cosine, and inverse tangent), you slide #135 to provide are ready to apply your knowledge and practice solving these additional assistance, when problems. needed. [This object is a pull tab]
Slide 133 / 252 Applications of Inverse Trigonometric Ratios A hockey player is 24 feet from the goal line. He shoots the puck directly at the goal. The height of the goal is 4 feet. What is the maximum angle of elevation at which the player can shoot the puck and still score a goal? 4 ft θ 24 ft Slide 134 / 252 Applications of Inverse Trigonometric Ratios tan θ = 4 4 ft 24 θ 24 ft ( ) θ = tan -1 4 24 θ = 9.46º The angle of elevation that the player can shoot the puck is a maximum of 9.46º. Slide 135 / 252 Applications of Inverse Trigonometric Ratios You lean a 20 foot ladder up against a wall. The base of the ladder is 5 feet from the edge of the wall. What is the angle of elevation is created by the ladder & the ground. 20 ft 5 ft
Slide 136 / 252 Applications of Inverse Trigonometric Ratios cos θ = 5 20 20 ft ( ) θ = cos -1 5 20 θ = 75.52º 5 ft Slide 137 / 252 37 Katherine looks down out of the crown of the statue of liberty to an incoming ferry about 345 feet. The distance from crown to the ground is about 250 feet. What is the angle of depression? Slide 137 (Answer) / 252 37 Katherine looks down out of the crown of the statue of liberty to an incoming ferry about 345 feet. The distance from crown to the ground is about 250 feet. crown What is the angle of depression? 345 ft 250 ft Answer ferry The angle of depression is about 46 degrees. [This object is a pull tab]
Slide 138 / 252 38 The Sear's Tower in Chicago, Illinois is 1451 feet tall. The sun is casting a 50 foot shadow on the ground. What is the angle of elevation created by the tip of the shadow and the ground? 1451 ft 50 ft Slide 138 (Answer) / 252 38 The Sear's Tower in Chicago, Illinois is 1451 feet tall. The sun is casting a 50 foot shadow on the ground. What is the angle of elevation created by the tip of the shadow and the ground? Answer θ = 88.03° 1451 ft [This object is a pull tab] 50 ft Slide 139 / 252 39 You lean a 30 foot ladder up against the side of your home to get into a bedroom on the second floor. The height of the window is 25 feet. What angle of elevation must you set the ladder at in order to reach the window? 30 ft 25 ft
Slide 139 (Answer) / 252 39 You lean a 30 foot ladder up against the side of your home to get into a bedroom on the second floor. The height of the window is 25 feet. What angle of elevation must you set the ladder at in order to reach the window? Answer θ = 56.44° 30 ft 25 ft [This object is a pull tab] Slide 140 / 252 40 You are looking out your bedroom window towards the tip of the shadow made by your home. Your friend measures the length of the shadow to be 10 feet long. If you are 20 feet off the ground, what is the angle of depression needed to see the tip of your home's shadow. Slide 140 (Answer) / 252 40 You are looking out your bedroom window towards the tip of the shadow made by your home. Your friend measures the length of the shadow to be 10 feet long. If you are 20 feet off the ground, what is the angle of depression needed to see the tip of your θ home's shadow. Answer 20 ft θ 10 ft θ = 63.43° [This object is a pull tab]
Slide 141 / 252 41 You return to view your home's shadow 3 hours later. Your friend measures the length of the shadow to be 25 feet long. If you are 20 feet off the ground, what is the angle of depression needed to see the tip of your home's shadow. Slide 141 (Answer) / 252 41 You return to view your home's shadow 3 hours later. Your friend measures the length of the shadow to be 25 feet long. If you are 20 feet off the ground, what is the angle of depression needed to see the tip of θ your home's shadow. Answer 20 ft θ 25 ft θ = 51.34° [This object is a pull tab] Slide 142 / 252 Review of the Pythagorean Theorem Return to the Table of Contents
Slide 143 / 252 Review of Pythagorean Theorem c 2 = a 2 + b 2 "c" is the hypotenuse "a" and "b" are the two legs; which leg is "a" and which is "b" doesn't matter. Slide 144 / 252 42 The legs of a right triangle are 7.0m and 3.0m, what is the length of the hypotenuse? Slide 144 (Answer) / 252 42 The legs of a right triangle are 7.0m and 3.0m, what is the length of the hypotenuse? Answer 7.6 [This object is a pull tab]
Slide 145 / 252 43 The legs of a right triangle are 2.0m and 12m, what is the length of the hypotenuse? Slide 145 (Answer) / 252 43 The legs of a right triangle are 2.0m and 12m, what is the length of the hypotenuse? Answer 12.2 [This object is a pull tab] Slide 146 / 252 44 The hypotenuse of a right triangle has a length of 4.0m and one of its legs has a length of 2.5m. What is the length of the other leg?
Slide 146 (Answer) / 252 44 The hypotenuse of a right triangle has a length of 4.0m and one of its legs has a length of 2.5m. What is the length of the other leg? Answer 3.1 [This object is a pull tab] Slide 147 / 252 45 The hypotenuse of a right triangle has a length of 9.0m and one of its legs has a length of 4.5m. What is the length of the other leg? Slide 147 (Answer) / 252 45 The hypotenuse of a right triangle has a length of 9.0m and one of its legs has a length of 4.5m. What is the length of the other leg? Answer 7.8 [This object is a pull tab]
Slide 148 / 252 46 What is the length of the third side? 7 4 Slide 148 (Answer) / 252 46 What is the length of the third side? Answer 8.1 7 [This object is a pull tab] 4 Slide 149 / 252 47 What is the length of the third side? 15 20
Slide 149 (Answer) / 252 47 What is the length of the third side? Answer 15 25 [This object is a pull tab] 20 Slide 150 / 252 48 What is the length of the third side? 7 4 Slide 150 (Answer) / 252 48 What is the length of the third side? Answer 5.7 7 [This object is a pull tab] 4
Slide 151 / 252 49 What is the length of the third side? 15 9 Slide 151 (Answer) / 252 49 What is the length of the third side? Answer 12 15 9 [This object is a pull tab] Slide 152 / 252 50 What is the length of the third side? 3 4
Slide 152 (Answer) / 252 50 What is the length of the third side? Answer 5 3 [This object is a pull tab] 4 Slide 153 / 252 Pythagorean Triples Triples are integer solutions of the Pythagorean Theorem. 5 3 3-4-5 is the most famous of the triples: You don't need a calculator if you recognize the sides are in this ratio. 4 Slide 154 / 252 51 What is the length of the third side? 6 8
Slide 154 (Answer) / 252 51 What is the length of the third side? Answer 10 6 Note: 6-8-10 = 2*(3-4-5) [This object is a pull tab] 8 Slide 155 / 252 52 What is the length of the third side? 20 12 Slide 155 (Answer) / 252 52 What is the length of the third side? Answer 16 20 12 Note: 12-16-20 = 4*(3-4-5) [This object is a pull tab]
Slide 156 / 252 53 (sinθ) 2 + (cosθ) 2 = ? 1 sinθ cosθ Slide 156 (Answer) / 252 53 (sinθ) 2 + (cosθ) 2 = ? Answer 1 2 = 1 1 sinθ [This object is a pull tab] cosθ Slide 157 / 252 54 Katherine looks down out of the crown of the statue of liberty to an incoming ferry about 345 feet. The distance from crown to the ground is about 250 feet. What is the distance from the ferry to the base of the statue?
Slide 157 (Answer) / 252 54 Katherine looks down out of the crown of the statue of liberty to an incoming ferry about 345 feet. The crown distance from crown to the ground is about 250 feet. What is the distance from the ferry to the base of 345 ft 250 ft the statue? x Answer base ferry of the statue [This object is a pull tab] The ferry is about 238 feet away from the statue. Slide 158 / 252 Converse of the Pythagorean Theorem Return to the Table of Contents Slide 159 / 252 Converse of the Pythagorean Theorem If the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle. B c If c 2 = a 2 + b 2 , then a ΔABC is a right triangle. A C b
Slide 160 / 252 Example Tell whether the triangle is a right triangle . Explain your reasoning. 24 E D 7 Remember c is the 25 longest side F Slide 160 (Answer) / 252 Example Tell whether the triangle is a right triangle . If c 2 = a 2 + b 2 , then ΔABC is a right Δ. Explain your reasoning. If 25 2 = 24 2 + 7 2 , then ΔABC is a right Δ. 625 = 576 + 49 24 E D 625 = 625 therefore ΔABC is a right Δ. Answer This example addresses MP2 & MP3 7 Remember c is the Additional Q's to address MP standards: 25 longest side What information do you have? (MP1) How do you determine if the triangle is a right F triangle? (MP3) Construct an equation to solve the problem (MP2) [This object is a pull tab] Slide 161 / 252 Theorem If the square of the longest side of a triangle is greater than the sum of the squares of the other two sides, then the triangle is obtuse. B c a If c 2 > a 2 + b 2 , then ΔABC is obtuse. A C b
Slide 162 / 252 Theorem If the square of the longest side of a triangle is less than the sum of the squares of the other two sides, then the triangle is acute. B c a If c 2 < a 2 + b 2 , then A ΔABC is acute. C b Slide 163 / 252 Example Classify the triangle as acute, right, or obtuse. Explain your reasoning. 15 13 17 Slide 163 (Answer) / 252 Example Classify the triangle as acute, right, or obtuse. Explain your reasoning. c = 17 17 2 ? 15 2 + 13 2 Answer 15 289 ? 225 + 169 13 289 < 394 Since c 2 < a 2 + b 2 the triangle is acute. 17 [This object is a pull tab]
Slide 164 / 252 55 Classify the triangle as acute, right, obtuse, or not a triangle. A acute 12 B right C obtuse 15 D not a triangle 11 Slide 164 (Answer) / 252 55 Classify the triangle as acute, right, obtuse, or not a triangle. A acute Answer 12 B right A C obtuse 15 D not a triangle 11 [This object is a pull tab] Slide 165 / 252 56 Classify the triangle as acute, right, obtuse, or not a triangle. A acute 4 B right 10 C obtuse 6 D not a triangle
Slide 165 (Answer) / 252 56 Classify the triangle as acute, right, obtuse, or not a triangle. A acute 4 Answer B right 10 D C obtuse D not a triangle 6 [This object is a pull tab] Slide 166 / 252 57 Classify the triangle as acute, right, obtuse, or not a triangle. A acute 5 3 B right C obtuse 6 D not a triangle Slide 166 (Answer) / 252 57 Classify the triangle as acute, right, obtuse, or not a triangle. A acute 5 3 B right Answer C C obtuse 6 D not a triangle [This object is a pull tab]
Slide 167 / 252 58 Classify the triangle as acute, right, obtuse, or not a triangle. 25 A acute B right 20 C obtuse D not a triangle 19 Slide 167 (Answer) / 252 58 Classify the triangle as acute, right, obtuse, or not a triangle. A acute 25 Answer B right A 20 C obtuse D not a triangle 19 [This object is a pull tab] Slide 168 / 252 59 Tell whether the lengths 35, 65, and 56 represent the sides of an acute, right, or obtuse triangle. A acute B right C obtuse
Slide 168 (Answer) / 252 59 Tell whether the lengths 35, 65, and 56 represent the sides of an acute, right, or obtuse triangle. A acute Answer B right A C obtuse [This object is a pull tab] Slide 169 / 252 60 Tell whether the lengths represent the sides of an acute, right, or obtuse triangle. acute triangle A B right triangle obtuse triangle C Slide 169 (Answer) / 252 60 Tell whether the lengths represent the sides of an acute, right, or obtuse triangle. acute triangle A Answer right triangle C B obtuse triangle C [This object is a pull tab]
Slide 170 / 252 Review If c 2 = a 2 + b 2 , then triangle is right. If c 2 > a 2 + b 2 , then triangle is obtuse. If c 2 < a 2 + b 2 , then triangle is acute. Slide 171 / 252 Special Right Triangles Return to the Table of Contents Slide 172 / 252 Special Right Triangles In this section you will learn about the properties of the two special right triangles. 45-45-90 30-60-90 45 o 60 o 90 o 30 o 90 o 45 o
Slide 173 / 252 Investigation: 45-45-90 Triangle Theorem Find the missing side lengths in the triangles. Leave answers in simplified radical/fractional form...NO DECIMALS! C y 1 2 45º 45º 1 2 Slide 173 (Answer) / 252 Investigation: 45-45-90 Triangle Theorem Find the missing side lengths in the triangles. Leave answers in simplified radical/fractional form...NO DECIMALS! 45º Answer y c = √2 2 y = 2√2 C y 1 2 2 45º 45º [This object is a pull tab] 1 2 Slide 174 / 252 Investigation: 45-45-90 Triangle Theorem Find the missing side lengths in the triangles. Leave answers in simplified radical/fractional form...NO DECIMALS! 45º W 3 C 4 45º 3 4
Slide 174 (Answer) / 252 Investigation: 45-45-90 Triangle Theorem Find the missing side lengths in the triangles. Leave answers in simplified radical/fractional form...NO DECIMALS! Answer w = 3√2 45º c = 4√2 W 3 C 4 45º [This object is a pull tab] 3 4 Slide 175 / 252 Investigation: 45-45-90 Triangle Theorem Find the missing side lengths in the triangles. Leave answers in simplified radical/fractional form...NO DECIMALS! C x 5 6 45º 45º 5 6 Slide 175 (Answer) / 252 Investigation: 45-45-90 Triangle Theorem Find the missing side lengths in the triangles. Leave answers in simplified radical/fractional form...NO DECIMALS! Answer c = 5√2 C x = 6√2 5 x 6 45º 45º [This object is a pull tab] 5 6
Slide 176 / 252 45-45-90 Triangle Theorem Using the side lengths that you found in the Investigation, can you figure out the rule, or formula, for the 45-45-90 Triangle Theorem? Slide 176 (Answer) / 252 45-45-90 Triangle Theorem A 45-45-90 triangle is an Using the side lengths that you found in the Investigation, isosceles right triangle, where can you figure out the rule, or formula, for the 45-45-90 the hypotenuse is √2 times the Triangle Theorem? length of the leg. Answer Math Practices: 45 o MP7 & MP8 x√2 x 45 o [This object is a pull tab] x Slide 177 / 252 45-45-90 Triangle Theorem This theorem can be proved algebraically using Pythagorean Theorem. a 2 + b 2 = c 2 45º x 2 + x 2 = c 2 x√2 2x 2 = c 2 x x√2 = c 45º x
Slide 178 / 252 45-45-90 Example Find the length of the missing sides. Write the answer in simplest radical form. 6 P Q 45º y x 45º R Slide 178 (Answer) / 252 45-45-90 Example By the Corollary to the Base Angles Thm, PQ=QR. y=6 Find the length of the missing sides. Write the answer in simplest radical form. Answer hypotenuse = √2(leg) x = √2(6) 6 P Q x = 6√2 45º y [This object is a pull tab] x 45º R Slide 179 / 252 45-45-90 Example Find the length of the missing sides. Write the answer in simplest radical form. y S T x 18 V
Slide 179 (Answer) / 252 45-45-90 Example Since, STU is an hypotenuse = leg isosceles rt triangle Find the length of the missing sides. Write the answer in simplest radical form. ST=TV x=y Answer y S T There are 2 ways to solve. x 18 [This object is a pull tab] V Slide 180 / 252 45-45-90 Example Find the length of the missing sides. x Write the answer in simplest radical form. y 8 Slide 180 (Answer) / 252 45-45-90 Example Find the length of the missing sides. x Write the answer in simplest radical form. Answer y 8 [This object is a pull tab]
Slide 181 / 252 61 Find the value of x. x A 5 B 5 √ 2 y 5 5 √ 2 C 2 Slide 181 (Answer) / 252 61 Find the value of x. x A 5 B 5 √ 2 y 5 5 √ 2 Answer C C 2 [This object is a pull tab] Slide 182 / 252 62 What is the length of the hypotenuse of an isosceles right √ 2 inches. triangle if the length of the legs is 8
Slide 182 (Answer) / 252 62 What is the length of the hypotenuse of an isosceles right √ 2 inches. triangle if the length of the legs is 8 hypotenuse = leg( ) Answer [This object is a pull tab] Slide 183 / 252 63 What is the length of each leg of an isosceles, if the length of the hypotenuse is 20 cm. Slide 183 (Answer) / 252 63 What is the length of each leg of an isosceles, if the length of the hypotenuse is 20 cm. hypotenuse = leg( ) Answer [This object is a pull tab]
Slide 184 / 252 Investigation: 30-60-90 Triangle Theorem Find the missing side lengths in the triangles. Leave answers in simplified radical/fractional form...NO DECIMALS! 30º 30º 2 z y 4 60º 60º 1 2 Slide 184 (Answer) / 252 Investigation: 30-60-90 Triangle Theorem Find the missing side lengths in the triangles. Leave answers in simplified radical/fractional form...NO DECIMALS! Answer z = √3 30º 30º y = 2√3 z 2 y 4 [This object is a pull tab] 60º 60º 1 2 Slide 185 / 252 Investigation: 30-60-90 Triangle Theorem Find the missing side lengths in the triangles. Leave answers in simplified radical/fractional form...NO DECIMALS! 30º 30º w v 6 8 60º 60º 3 4
Slide 185 (Answer) / 252 Investigation: 30-60-90 Triangle Theorem Find the missing side lengths in the triangles. Leave answers in simplified radical/fractional form...NO DECIMALS! Answer 30º 30º w = 3√3 c = 4√3 w v 6 8 60º 60º [This object is a pull tab] 3 4 Slide 186 / 252 Investigation: 30-60-90 Triangle Theorem Find the missing side lengths in the triangles. Leave answers in simplified radical/fractional form...NO DECIMALS! 30º 30º u t 10 12 60º 60º 5 6 Slide 186 (Answer) / 252 Investigation: 30-60-90 Triangle Theorem Find the missing side lengths in the triangles. Leave answers in simplified radical/fractional form...NO DECIMALS! Answer u = 5√3 t = 6√3 30º 30º u t 10 12 [This object is a pull tab] 60º 60º 5 6
Slide 187 / 252 30-60-90 Triangle Theorem Using the side lengths that you found in the Investigation, can you figure out the rule, or formula, for the 30-60-90 Triangle Theorem? Slide 187 (Answer) / 252 30-60-90 Triangle Theorem In a 30-60-90 right triangle, Using the side lengths that you found in the Investigation, the hypotenuse is twice the can you figure out the rule, or formula, for the 30-60-90 length of the shorter leg Triangle Theorem? and the longer leg is Answer √3 times the length of the shorter leg. Math Practices: 60 o 2x MP7 & MP8 x 30 o [This object is a pull tab] x # 3 Slide 188 / 252 30-60-90 Triangle Theorem This theorem can be proved using 2x 60 o x an equilateral triangle and Pythagorean Theorem. 30 o x √ 3 For right triangle ABD, BD is a perpendicular bisector. B let a = x, c = 2x and b = BD a 2 + b 2 = c 2 30º 30º x 2 + b 2 = (2x) 2 c=2x 2x x 2 + b 2 = 4x 2 b b 2 = 3x 2 b = x√3 60º 60º A C a=x D x
Slide 189 / 252 30-60-90 Example Example: Find the length of the missing sides G of the right triangle. 30º y x 60º H F 5 Slide 190 / 252 30-60-90 Example G Recall triangle inequality, the shortest side is opposite the smallest angle and the longest 30º side is opposite the largest angle. y x HF is the shortest side GF is the longest side (hypotenuse) GH is the 2nd longest side HF < GH < GF 60º H F 5 Slide 190 (Answer) / 252 30-60-90 Example G hypotenuse = 2(shorter leg) x = 2(5) Recall triangle inequality, the shortest side is x = 10 opposite the smallest angle and the longest 30º side is opposite the largest angle. Answer longer leg = √3(shorter leg) y = √3(5) y x HF is the shortest side y = 5√3 GF is the longest side (hypotenuse) GH is the 2nd longest side HF < GH < GF 60º H F 5 [This object is a pull tab]
Slide 191 / 252 30-60-90 Example x M A Example: 60º Find the length of the missing sides of the right triangle. 9 y 30º T Slide 191 (Answer) / 252 30-60-90 Example x M A MA is the shorter leg Example: and MT is the longer leg 60º Find the length of the missing sides Answer longer leg = √3(shorter leg) of the right triangle. 9 = √3(x) 3√3 = x 9 y 30º [This object is a pull tab] T Slide 192 / 252 30-60-90 Example Example: Find the length of the missing sides of the right triangle. 15 30º x y 60º
Slide 192 (Answer) / 252 30-60-90 Example Example: Find the length of the missing sides of the right triangle. Answer 15 30º x y 60º [This object is a pull tab] Slide 193 / 252 30-60-90 Example Example: Find the area of the triangle. 14 ft Slide 193 (Answer) / 252 30-60-90 Example Example: These next 2 example (slides #207 - Find the area of the triangle. 209) address MP1, MP2, MP4, MP5. 14 ft Additional Q's to address MP's: Math Practice What information do you have? (MP1) What side length(s) can you find? (MP1) What should we add to our diagram to solve this problem? (MP4 & MP5) Create an equation for this problem. (MP2) [This object is a pull tab]
Slide 194 / 252 30-60-90 Example The altitude (or height) divides the triangle into two 14 ft h 30 o -60 o -90 o triangles. ? ? The length of the shorter leg is 7 ft. The length of the longer leg is 7√3 ft. A = 1 / 2 b(h) = 1 / 2 14(7√3) A = 49√3 square ft ≈ 84.87 square ft Slide 195 / 252 30-60-90 Example Example: Find the area of the triangle. 9 ft 30 o Slide 195 (Answer) / 252 30-60-90 Example Example: short leg = 4.5 ft Find the area of the triangle. long leg = 4.5 ft Answer 9 ft 30 o [This object is a pull tab]
Slide 196 / 252 64 Find the value of x. A 7 60º 7 B 7 √ 3 7 √ 2 30º C x 2 D 14 Slide 196 (Answer) / 252 64 Find the value of x. A 7 60º Answer 7 B 7 √ 3 B 7 √ 2 30º C 2 x D 14 [This object is a pull tab] Slide 197 / 252 65 Find the value of x. A 7 x B 7 √ 3 7 √ 2 7 √ 2 C 2 D 14
Slide 197 (Answer) / 252 65 Find the value of x. A 7 x B 7 √ 3 7 √ 2 Answer 7 √ 2 D C 2 D 14 [This object is a pull tab] Slide 198 / 252 66 Find the value of x. 7√3 A 7 30 o x B 7 √ 3 7 √ 2 60 o C 2 D 14 Slide 198 (Answer) / 252 66 Find the value of x. 7√3 A 7 30 o x B 7 √ 3 Answer A 7 √ 2 60 o C 2 D 14 [This object is a pull tab]
Slide 199 / 252 67 The hypotenuse of a 30º-60º-90º triangle is 13 cm. What is the length of the shorter leg? Slide 199 (Answer) / 252 67 The hypotenuse of a 30º-60º-90º triangle is 13 cm. What is the length of the shorter leg? Answer shorter leg = 13/2 shorter leg = 6.5cm [This object is a pull tab] Slide 200 / 252 68 The length the longer leg of a 30º-60º-90º triangle is 7 cm. What is the length of the hypotenuse?
Slide 200 (Answer) / 252 68 The length the longer leg of a 30º-60º-90º triangle is 7 cm. What is the length of the hypotenuse? shorter leg = Answer hypotenuse = = [This object is a pull tab] Slide 201 / 252 Real World Example The wheelchair ramp at your school has a height of 2.5 feet and rises at angle of 30º. What is the length of the ramp? Slide 201 (Answer) / 252 Real World Example This example (this and the next 2 slides) addresses MP1, MP2, MP4 & MP5. Math Practice Additional Q's to address MP's: What information do you have? (MP1) What side length(s) can you find? (MP1) What should we add to our diagram to solve this problem? (MP4 & MP5) Create an equation for this problem. The wheelchair ramp at your school has a height of 2.5 feet and (MP2) [This object is a pull tab] rises at angle of 30º. What is the length of the ramp?
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