Slide 1 / 162 Slide 2 / 162 Algebra II Trigonometric Functions 2015-12-17 www.njctl.org Slide 3 / 162 Slide 4 / 162 click on the topic to go Trig Functions to that section Radians & Degrees Radians & Degrees & Co-terminal angles · Arc Length & Area of a Sector · and Co-Terminal Unit Circle · Angles Graphing · Trigonometric Identities Return to · Table of Contents Slide 5 / 162 Slide 6 / 162 Radians and Degrees A few definitions: A central angle of a circle is an angle whose vertex is the center of the circle. One radian is the measure An intercepted arc is the part of the circle that of a central angle that includes the points of intersection with the central intercepts an arc whose angle and all the points in the interior of the angle. length is equal to the radius of the circle. There are , or a little intercepted arc more than 6, radians in a circle. central angle Click on the circle for an animated view of radians.
Slide 7 / 162 Slide 8 / 162 Converting from Degrees to Radians Converting from Radians to Degrees There are 360 in a circle. Therefore 2 radians = 360 ∘ 360 ∘ = 2 radians 360 180 1 radian = = degrees 2 2 radians 1 ∘ = = 360 180 Use this conversion factor to covert radians to degrees. Use this conversion factor to covert degrees to radians. Example: Convert 50 ∘ and 90 ∘ to radians. Example: Convert 50 ∘ and 90 ∘ to radians. Example: Convert and to radians 4 5 5 50 ∘ ⋅ 50 ∘ 18 radians radians 180 = = 180 ⋅ = 45 ∘ 180 18 4 ⋅ 180 = 180 ∘ 90 ∘ ⋅ 90 ∘ 2 radians radians 180 = = 180 2 Slide 9 / 162 Slide 10 / 162 Converting between Radians and Degrees Converting between Radians and Degrees Convert radians to degrees Convert degrees to radians radians radians radians Slide 11 / 162 Slide 12 / 162
Slide 13 / 162 Slide 14 / 162 4 Convert radians to degrees: Slide 15 / 162 Slide 16 / 162 Angles Negative Angle - terminal Positive Angle - terminal side rotates in a side rotates in a counter- Terminal side Terminal side clockwise direction clockwise direction Initial side Initial side α = - 37 ∘ 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 17 / 162 Slide 18 / 162 Coterminal Angles Drawing angles in standard position 310 ∘ 500 ∘ 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 ∘ . 40 ∘ 435 ∘ Each quadrant is 90 ∘ , 500 ∘ is 140 ∘ more than -285 ∘ 75 + 360 = 435 75 ∘ and 310 ∘ is 40 ∘ more 360 ∘ , so the angle makes a complete revolution 75 - 360 = -285 than 270 ∘ , so the counterclockwise and terminal side is 40 ∘ past then another 140 ∘ . the negative y-axis.
Slide 19 / 162 Slide 20 / 162 5 Which angles are coterminal with 40 ∘ ? (Select all 6 Which graph represents 425 ∘ ? that are correct.) B A A 320 B -320 C 400 D -400 C D Slide 21 / 162 Slide 22 / 162 7 Which graph represents ? 8 Which angle is NOT coterminal with -55 ∘ ? A B A 305 ∘ B 665 ∘ C -415 ∘ D -305 ∘ D C Slide 23 / 162 Slide 24 / 162 9 Which angle is coterminal with ? A Arc Length & Area of a Sector B C Return to D Table of Contents
Slide 25 / 162 Slide 26 / 162 Who is getting more pie? Who is getting Arc length and the area of a sector more of the crust at the outer edge? (Measured in radians) 40 ∘ 45 ∘ r arc length s Emily's slice is cut Chester's slice is cut from a 9 inch pie. from an 8 inch pie. sector Arc length: s = r Arc length: s = r (Assume both pies are the same height.) Area of sector: A = Area of sector: A = How do these formulas relate to the area and the (Try to work this out in your groups. The solution is on the next slide) circumference of a circle? Slide 27 / 162 Slide 28 / 162 10 What is the top surface area of this slice of pizza 40 ∘ from an 18-inch pie? 45 ∘ click click The top of Emily's The top of Chester's piece has an area of piece has an area of 45 ∘ Emily's crust has a Chester's crust has a length of length of Slide 29 / 162 Slide 30 / 162 11 What is the arc length of the outer edge of 12 If the radius of this circular saw blade is 10 this slice of pizza from an 18-inch pie? inches and there are 40 teeth on the blade, how far apart are the tips of the teeth? 45 ∘
Slide 31 / 162 Slide 32 / 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? Unit Circle i n 4 8 inches Return to Table of Contents Slide 33 / 162 Slide 34 / 162 The unit circle allows us to extend trigonometry The Unit Circle beyond angles of triangles to angles of all measures. The circle x 2 + y 2 = 1 , with center (0,0) and radius 1, (0,1) In this triangle, is called the unit circle . (a,b) b sin # = = b 1 1 b (0,1) Quadrant II : x is a Quadrant I : x and y (-1,0) θ cos # = = a negative and y is a 1 are both positive (1,0) positive 1 so the coordinates of (a,b) are also (-1,0) (1,0) (0,-1) (cos # , sin # ) Quadrant IV : x is Quadrant III : x and y positive and y is are both negative negative For any angle in standard position, the point where (0,-1) the terminal side of the angle intercepts the circle is called the terminal point. Slide 35 / 162 Slide 36 / 162 In this example, the terminal point is in Quadrant IV. If we look at the triangle, we can see that Click the star below to go to the Khan Academy Unit Circle Manipulative try some problems: 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 37 / 162 Slide 38 / 162 The Tangent Function What are the coordinates of point C? Recall SOH-CAH-TOA In this example, we know the angle. Using a calculator, we find that cos 44 ∘ ≈ .72 and sin 44 ∘ ≈ .69, sin = hypotenuse opp # opposite side so the coordinates of C are approximately (0.72, 0.69). hyp cos = adj # hyp tan = # opp # 1 adjacent side adj It is also true that tan = . # sin cos # # opp Why? hyp = ⋅ = = tan opp hyp opp # adj adj hyp adj Note that 0.72 2 + 0.69 2 ≈ 1! hyp Slide 39 / 162 Slide 40 / 162 Angles in the Unit Circle Example: Given a terminal point , find # , tan # and csc # . Example: Given a terminal point on the unit circle To find #, use sin -1 or cos -1 : (- ). sin -1 ( ) = # Find the value of cos, sin and tan of the angle. # # 28.1 ∘ Solution: Let the angle be . tan# = sin#/ cos# x = cos , so cos = . tan # = y = sin , so sin = . csc# = 1/ sin# ⋅ tan = = = = csc # = Note the "hidden" Pythagorean Triple, 8, 15, 17). (Shortcut: Just cross out the 41's in the complex fraction.) Slide 41 / 162 Slide 42 / 162 Example: Find the x-value of point A, θ and the tan θ. Example: Given the terminal point of ( -5 / 13 , -12 / 13 ). F ind sin x, cos x, and tan x. For every point on the circle, θ 22.3 -22.3 A 5 (__, - ) 13 Since x is in quadrant III, x = - 12 13 5 sin -1 (- ) ≈ -22.3 , BUT θ is in quadrant III, so θ = 180 + 22.3 = 13 202.3 (notice how 202.3 and -22.3 have the same sine) tan θ = = = 5 sin θ cos θ 12
Slide 43 / 162 Slide 44 / 162 14 What is tan θ? 15 What is sin θ ? 3 (- ,__) 3 5 (- ,__) 5 A A θ θ B B C C D D Slide 45 / 162 Slide 46 / 162 16 What is θ (give your answer to the nearest degree) ? 17 Given the terminal point , find tan x. 3 (- ,__) 5 θ Slide 47 / 162 Slide 48 / 162 Equilateral and isosceles triangles occur frequently in geometry and trigonometry. The angles in these triangles are multiples of 30 and 45 . A calculator will give approximate values for the trig functions of these angles, but we often need to know the exact values. Isosceles Right Triangle Equilateral Triangle (the altitude divides the triangle into two 30-60-90 triangles)
Slide 49 / 162 Slide 50 / 162 Special Right Triangles Special Triangles and the Unit Circle ( , ) (- , ) 1 1 45 ∘ 45 ∘ - Multiples of 45 angles have sin and cos of ± , depending on the quadrant. (see Triangle Trig Review unit for more detail on this topic) Slide 51 / 162 Slide 52 / 162 Drag the degree and radian angle measures to the angles of the circle: # # 5# 7# 3# 3# 60 o # 60 o 2# 0 4 2 4 2 4 4 45 o 45 o 0 ∘ 45 ∘ 30 o 30 o 90 ∘ 135 ∘ 180 ∘ 30 o 30 o 225 ∘ 270 ∘ 45 o 45 o 315 ∘ 60 o 60 o 360 ∘ Slide 53 / 162 Slide 54 / 162 Fill in the coordinates of x and y for each point Special Triangles and the Unit Circle on the unit circle: ( , ) ( , ) # ( , ) 2 ( , ) 3# 1 ( , ) # 1 4 4 30 ∘ 60 ∘ ( , ) 0 ( , ) 0 # 2# 1 5# 4 -1 7# 3# Angles that are multiples of 30 have sin and cos ( , ) 2 4 ( , ) of ± and ± . ( , )
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