Slide 1 / 92 Slide 2 / 92 Algebra II Trigonometry of the Triangle 2015-04-21 www.njctl.org Slide 3 / 92 Slide 4 / 92 click on the topic to go Trig Functions to that section Trigonometry of the Right Triangle · Trigonometry of Inverse Trig Functions · the Right Triangle Problem Solving with Trig · Special Right Triangles · Law of Sines · Return to Table of Contents Law of Cosines · Slide 5 / 92 Slide 6 / 92 Recall the Right Triangle Trigonometry means "measurement of triangles". In its earliest applications, it dealt with triangles and the A relationships between the lengths of their sides and hypotenuse the angles between those sides. leg Historically trig was used for astronomy and geography, but it has been used for centuries in many C B leg other fields. Today, among many other fields, it has The sum of the measures of the angles is 180 ∘ . applications in music, financial market analysis, electronics, probability, biology, medicine, The hypotenuse is the longest side and opposite the architecture, economics, engineering and game right angle. The other two sides are called legs. development. In any right triangle, the Pythagorean Theorem tell us that: leg 2 + leg 2 = hypotenuse 2 .
Slide 7 / 92 Slide 8 / 92 Similar Triangles Pythagorean Triples (these are helpful to know) If the two acute angles of two right triangles are congruent, then the triangles are similar and the A "Pythagorean Triple" is a set of whole numbers, a, b and c that fits the rule: sides are proportional. a 2 + b 2 = c 2 c f a d Recognizing these numbers can save time and effort b in solving trig problems. e Here are the first few: 3, 4, 5 5, 12, 13 7, 24, 25 The ratios of the sides are the trig ratios. 8, 15, 17 9, 40, 41 11, 60, 61 Also, any multiple of a triple is another triple: 6, 8, 10 10, 24, 26 and so on Slide 9 / 92 Slide 10 / 92 Trigonometric Ratios A D The fundamental trig ratios are: Sine abbreviated as "sin" (pronounced like "sign") E C B F Cosine abbreviated as "cos", but If ∆ ABC ∼ ∆DEF , drag the measurements into the pronounced "cosine" proportions: Tangent abbreviated as "tan", but pronounced "tangent" AB = AC = DF AB = DF AC Greek letters like θ, "theta", and , " beta ", # are often used to represent angles. Upper- EF DE BC case letters are also used. BC EF DE sin θ means "the sine of the angle θ" cos θ means "the cosine of the angle θ" tan θ means "the tangent of the angle θ" Slide 11 / 92 Slide 12 / 92 Trigonometric Ratios In order to name the trig ratios, Trigonometric Ratios you need a reference angle, θ . Notice what happens If θ is the reference angle, θ when θ is the other then hypotenuse angle. hypotenuse - the leg opposite θ is called adjacent opposite side side If the other angle is our the opposite side. (You have reference angle θ, then to cross the triangle to get to the opposite side.) the sides labeled as θ opposite and adjacent opposite side adjacent side - the adjacent side is one of switch places. the sides of θ, but not the hypotenuse The hypotenuse is always the hypotenuse. - the side opposite the right angle is the hypotenuse
Slide 13 / 92 Slide 14 / 92 Trigonometric Ratios 1 sin θ opp opposite side sin θ = = hyp hypotenuse θ θ hypotenuse adjacent side adj 16 cos θ = = 7 hypotenuse hyp adjacent side tan θ = opposite side adjacent side = opp adj opposite side 14 SOH-CAH-TOA: use this acronym to remember the trig ratios For any right triangle with angle # the ratios will be equal. Slide 15 / 92 Slide 16 / 92 2 cos θ 3 tan θ θ 16 7 16 7 θ 14 14 Slide 17 / 92 Slide 18 / 92 Reciprocal Trig Functions 4 cos # = There are three more ratios that can be created comparing the sides of the triangle, cosecant (csc), secant (sec), and cotangent (cot): 1 hypotenuse hyp θ csc θ = = = sin θ opposite opp 16 hypotenuse 7 adjacent 1 hypotenuse hyp side sec θ = = = cos θ adjacent adj θ opposite side 14 adjacent = opp 1 adj cot θ = = tan θ opposite
Slide 19 / 92 Slide 20 / 92 Evaluating Trig Functions 5 sec # = Example: Find the values of the six trig functions of θ in the triangle below. Solution: Use the Pythagorean Theorem to θ # find the missing side: 3 2 + 4 2 = c 2 , so c = 5. c 13 5 3 5 4 sin θ = csc θ = 4 5 5 3 cos θ = sec θ = 4 5 3 3 4 tan θ = cot θ = 12 3 4 Slide 21 / 92 Slide 22 / 92 7 cot # = 6 sin # = (answer in decimal form) θ 8.0 3.0 8.5 3.0 θ 8.5 8.0 Slide 23 / 92 Slide 24 / 92 9 csc # = 8 cot # = θ 16 16 7 7 θ 14 14
Slide 25 / 92 Slide 26 / 92 Using Trig Ratios 10 sec # = If you know the length of a side and the measure of one of the acute angles in a right triangle, you can use trig ratios to find the other sides. 16 7 θ 14 Slide 27 / 92 Slide 28 / 92 Trigonometric Ratios Trigonometric Ratios For example, let's find the sin 30 ∘ is always equal to the sin 30 ∘ = x length of side x. 7 same number, regardless of 7sin 30 ∘ = x the size of the triangle. To The side we're looking for is find the value of sin 30 ∘ , we opposite the given angle; 7 can use a calculator that has x 7 and the given length is the x trig functions. hypotenuse; sin 30 ∘ = 0.5, so we'll use the trig function 30 o 30 o that relates these two: so x = 7(0.5) = 3.5 opposite side opp sinθ = = hypotenuse hyp NOTE: Be sure your calculator is set to degree mode. (continued on next slide) Slide 29 / 92 Slide 30 / 92 Trigonometric Ratios Trigonometric Ratios Example 3: Find x. Example 2: Find x. We are looking for the The side we're looking for is opposite side, and are adjacent to the given angle 50 o given the adjacent side. and the given length is the 9 9 The trig function that hypotenuse relates these is tangent: tan θ = opp so we'll use the trig function adj that relates these two: 25 o x adj x x cosθ = hyp click to reveal tan 50 ∘ = 9 cos 25 ∘ = x x = 9tan 50 ∘ ≈ 10.7 click to reveal 9 x = 9cos 25 ∘ ≈ 8.16
Slide 31 / 92 Slide 32 / 92 Example: Find x. x 11 x = ? 12 We are looking for the 22 ∘ opposite side, and are given the hypotenuse. The trig function that This time the x is on the relates these is sine: 35 bottom. To solve we sin θ = opp would multiply both hyp sides by x and then sin 22 ∘ = 12 divide by sin 22 ∘ . x Enter this (Remember this short 64 o 12 into the x = cut: switch the x and sin 22 ∘ x calculator the sin 22 ∘ .) x ≈ 32 Slide 33 / 92 Slide 34 / 92 12 x = ? 13 x = ? x 28 44 o x 28 36 o Slide 35 / 92 Slide 36 / 92 14 x = ? 7.4 37 o Inverse Trig Functions x Return to Table of Contents
Slide 37 / 92 Slide 38 / 92 If we know the lengths of two sides of a right Note: In the next unit we will explore the values of triangle, we can use inverse trig functions to find the angles. trig functions for any angle. At that point, it will be clear that because the sin, cos and tan functions "arcsin(some number)" is equal to the angle repeat (they are not one-to-one), their inverses are whose sine is (some number). not functions. If we restrict the domain, however, arcsin is often written as sin -1 the functions are one-to-one and their inverses are functions. When given a trig function value, we use a calculator to find the angle measure. Use the When the inverse function is entered into the calculator, the response is a number in the restricted and keys to calculate sin -1 . interval. Use inverse trig functions when you need to find the angle . Slide 39 / 92 Slide 40 / 92 Find the value of the angles and other side of 8 Example: In this triangle, tan θ = . 15 the triangle. 1) Use the Pythagorean Theorem to find third side. We are looking for the angle 8 (don't forget to think about whose tangent is . 15 8 Pythagorean triples) 9 tan -1 ( ) ≈ 28.1 ∘ 2) Use any inverse trig θ function to find one of the 15 12 angles. 3) Subtract that angle measure from 90 ∘ to find (Enter " (8 ÷ 15)" into the calculator.) The third side is 15. click the other angle. ≈ 53.1 ∘ ≈ 36.9 ∘ Slide 41 / 92 Slide 42 / 92 15 Find the value of the angle indicated. 16 Find the value of the angle indicated. 16 64 23 24
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