Algebra II Trigonometric Functions 2015-12-17 www.njctl.org Slide - PDF document

Slide 1 / 162 Slide 2 / 162 Algebra II Trigonometric Functions 2015-12-17 www.njctl.org Slide 3 / 162 click on the topic to go Trig Functions to that section Radians & Degrees & Co-terminal angles · Arc Length & Area of a Sector · Unit Circle · Graphing · Trigonometric Identities ·

Slide 4 / 162 Radians & Degrees and Co-Terminal Angles Return to Table of Contents Slide 5 / 162 A few definitions: A central angle of a circle is an angle whose vertex is the center of the circle. An intercepted arc is the part of the circle that includes the points of intersection with the central angle and all the points in the interior of the angle. intercepted arc central angle Slide 6 / 162 Radians and Degrees One radian is the measure of a central angle that intercepts an arc whose length is equal to the radius of the circle. There are , or a little more than 6, radians in a circle. Click on the circle for an animated view of radians.

Slide 7 / 162 Converting from Degrees to Radians There are 360 in a circle. Therefore 360 ∘ = 2 radians 2 radians 1 ∘ = = 360 180 Use this conversion factor to covert degrees to radians. Example: Convert 50 ∘ and 90 ∘ to radians. Example: Convert 50 ∘ and 90 ∘ to radians. 5 5 50 ∘ ⋅ 50 ∘ 18 radians radians 180 = = 180 18 90 ∘ 90 ∘ ⋅ 2 radians radians 180 = = 180 2 Slide 8 / 162 Converting from Radians to Degrees 2 radians = 360 ∘ 360 180 1 radian = = degrees 2 Use this conversion factor to covert radians to degrees. Example: Convert and to radians 4 180 ⋅ = 45 ∘ 4 ⋅ 180 = 180 ∘ Slide 9 / 162 Converting between Radians and Degrees Convert degrees to radians

Slide 9 (Answer) / 162 Converting between Radians and Degrees Convert degrees to radians Answer [This object is a pull tab] Slide 10 / 162 Converting between Radians and Degrees Convert radians to degrees radians radians radians Slide 10 (Answer) / 162 Converting between Radians and Degrees Convert radians to degrees radians Answer radians radians [This object is a pull tab]

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Slide 14 / 162 4 Convert radians to degrees: Slide 14 (Answer) / 162 4 Convert radians to degrees: Answer 67.5 0 [This object is a pull tab] Slide 15 / 162 Angles Terminal side Terminal side Initial side Initial side Angle Angle in standard position An angle is formed by rotating a ray about its endpoint. The starting position is the initial side and the ending position is the terminal side. When, on the coordinate plane, the vertex of the angle is the origin and the initial side is the positive x-axis, the angle is in standard position.

Slide 16 / 162 Negative Angle - terminal Positive Angle - terminal side rotates in a side rotates in a counter- clockwise direction clockwise direction α = - 37 ∘ Slide 17 / 162 Drawing angles in standard position 310 ∘ 500 ∘ 40 ∘ Each quadrant is 90 ∘ , 500 ∘ is 140 ∘ more than 360 ∘ , so the angle makes and 310 ∘ is 40 ∘ more a complete revolution than 270 ∘ , so the counterclockwise and terminal side is 40 ∘ past then another 140 ∘ . the negative y-axis. Slide 18 / 162 Coterminal Angles Angles that have the same terminating side are coterminal. To find coterminal angles add or subtract multiples of 360 ∘ for degrees and 2 for radians. Example: Find one positive and one negative angle that are terminal with 75 ∘ . 435 ∘ -285 ∘ 75 + 360 = 435 75 ∘ 75 - 360 = -285

Slide 19 / 162 5 Which angles are coterminal with 40 ∘ ? (Select all that are correct.) A 320 B -320 C 400 D -400 Slide 19 (Answer) / 162 5 Which angles are coterminal with 40 ∘ ? (Select all that are correct.) A 320 B -320 C 400 Answer B, C D -400 [This object is a pull tab] Slide 20 / 162 6 Which graph represents 425 ∘ ? B A C D

Slide 20 (Answer) / 162 6 Which graph represents 425 ∘ ? B A Answer A C D [This object is a pull tab] Slide 21 / 162 7 Which graph represents ? B A D C Slide 21 (Answer) / 162 7 Which graph represents ? A B Answer D [This object is a pull tab] D C

Slide 22 / 162 8 Which angle is NOT coterminal with -55 ∘ ? A 305 ∘ B 665 ∘ C -415 ∘ D -305 ∘ Slide 22 (Answer) / 162 8 Which angle is NOT coterminal with -55 ∘ ? A 305 ∘ B 665 ∘ Answer C -415 ∘ A, B, C D -305 ∘ [This object is a pull tab] Slide 23 / 162 9 Which angle is coterminal with ? A B C D

Slide 23 (Answer) / 162 9 Which angle is coterminal with ? A B Answer B C [This object is a pull tab] D Slide 24 / 162 Arc Length & Area of a Sector Return to Table of Contents Slide 25 / 162 Arc length and the area of a sector (Measured in radians) r arc length s sector Arc length: s = r Arc length: s = r Area of sector: A = Area of sector: A = How do these formulas relate to the area and the circumference of a circle?

Slide 26 / 162 Who is getting more pie? Who is getting more of the crust at the outer edge? 40 ∘ 45 ∘ Emily's slice is cut Chester's slice is cut from a 9 inch pie. from an 8 inch pie. (Assume both pies are the same height.) (Try to work this out in your groups. The solution is on the next slide) Slide 27 / 162 40 ∘ 45 ∘ click click The top of Emily's The top of Chester's piece has an area of piece has an area of Emily's crust has a Chester's crust has a length of length of Slide 28 / 162 10 What is the top surface area of this slice of pizza from an 18-inch pie? 45 ∘

Slide 28 (Answer) / 162 10 What is the top surface area of this slice of pizza from an 18-inch pie? 45 ∘ Answer 1 81 2 ⋅ 9 2 ⋅ 4 ≈ 31.8 in 2 = 8 [This object is a pull tab] Slide 29 / 162 11 What is the arc length of the outer edge of this slice of pizza from an 18-inch pie? 45 ∘ Slide 29 (Answer) / 162 11 What is the arc length of the outer edge of this slice of pizza from an 18-inch pie? Answer 9 ⋅ 4 ≈ 7.1 in 45 ∘ [This object is a pull tab]

Slide 30 / 162 12 If the radius of this circular saw blade is 10 inches and there are 40 teeth on the blade, how far apart are the tips of the teeth? Slide 30 (Answer) / 162 12 If the radius of this circular saw blade is 10 inches and there are 40 teeth on the blade, how far apart are the tips of the teeth? 2 Answer ⋅ 10 ≈ 1.6 inches 40 [This object is a pull tab] Slide 31 / 162 13 Challenge Question: Given a dart board as shown. If a dart thrown randomly lands somewhere on the board, what is the probability that it will land on a red region? i n 4 8 inches

Slide 31 (Answer) / 162 13 Challenge Question: Given a dart board as shown. If a dart thrown randomly lands somewhere on the Area of each of the inner red regions: board, what is the probability that it will land on a red region? Area of each of the outer red regions: n 4 i 8 inches Total area of all red regions: Area of the entire dart board: Probability: or about 10% [This object is a pull tab] Slide 32 / 162 Unit Circle Return to Table of Contents Slide 33 / 162 The Unit Circle The circle x 2 + y 2 = 1 , with center (0,0) and radius 1, is called the unit circle . (0,1) Quadrant II : x is Quadrant I : x and y negative and y is are both positive positive 1 (-1,0) (1,0) Quadrant IV : x is Quadrant III : x and y positive and y is are both negative negative (0,-1)

Slide 34 / 162 The unit circle allows us to extend trigonometry beyond angles of triangles to angles of all measures. (0,1) In this triangle, (a,b) b sin # = = b 1 1 b a (-1,0) θ cos # = = a a 1 (1,0) so the coordinates of (a,b) are also (0,-1) (cos # , sin # ) For any angle in standard position, the point where the terminal side of the angle intercepts the circle is called the terminal point. Slide 35 / 162 In this example, the terminal point is in Quadrant IV. If we look at the triangle, we can see that sin(-55 ) = 0.82 cos(-55 ) = 0.57 0.57 EXCEPT that we have to take -55 the direction into account, 0.82 1 and so sin(-55 ) is negative because the y value is below the x-axis. For any angle θ in standard position, the terminal point has coordinates (cos θ, sinθ). Slide 36 / 162 Click the star below to go to the Khan Academy Unit Circle Manipulative try some problems:

Slide 37 / 162 What are the coordinates of point C? In this example, we know the angle. Using a calculator, we find that cos 44 ∘ ≈ .72 and sin 44 ∘ ≈ .69, so the coordinates of C are approximately (0.72, 0.69). 1 Note that 0.72 2 + 0.69 2 ≈ 1! Slide 38 / 162 The Tangent Function Recall SOH-CAH-TOA sin = hypotenuse opp # opposite side hyp cos = adj # hyp tan = # opp # adjacent side adj It is also true that tan = . # sin cos # # opp Why? hyp = ⋅ = = tan opp hyp opp # adj adj hyp adj hyp Slide 39 / 162 Angles in the Unit Circle Example: Given a terminal point on the unit circle (- ). Find the value of cos, sin and tan of the angle. Solution: Let the angle be . x = cos , so cos = . y = sin , so sin = . ⋅ tan = = = = (Shortcut: Just cross out the 41's in the complex fraction.)