Pre-Calc. Slide 1 / 207. Slide 2 / 207. Slide 3 / 207. Trigonometry. Table of Contents
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1 Slide 1 / 207 Pre-Calc Slide 2 / 207 Trigonometry Table of Contents Slide 3 / 207 Unit Circle Graphing Law of Sines Law of Cosines Pythagorean Identities Angle Sum/Difference Double Angle Half Angle Power Reducing Sum to Product Product to Sum Inverse Trig Functions Trig Equations click on the topic to go to that section
2 Slide 4 / 207 Unit Circle Return to Table of Contents Unit Circle Slide 5 / 207 Goals and Objectives Students will understand how to use the Unit Circle to find angles and determine their trigonometric value. Unit Circle Slide 6 / 207 Why do we need this? The Unit Circle is a tool that allows us to determine the location of any angle.
3 Unit Circle Slide 7 / 207 Special Right Triangles Unit Circle Slide 8 / 207 Example 1: Find a Example 2: Find b & c 6 a 4 c b Unit Circle Slide 9 / 207 Example 3: Find d d 8 Example 4: Find e 9 e
4 Unit Circle Slide 10 / 207 Example 5: Find f Example 6: Find g & h f 1 g h 1 Unit Circle Slide 11 / o 45 o 60 o 60 o 45 o 30 o 30 o 30 o 45 o 60 o 60 o 45 o Unit Circle Slide 12 / 207
5 Unit Circle Slide 13 / 207 Unit Circle Slide 14 / 207 Unit Circle Slide 15 / 207
6 Slide 16 / 207 Slide 17 / 207 Slide 18 / 207
7 Slide 19 / 207 Slide 20 / 207 Slide 21 / 207
8 Unit Circle Slide 22 / Which function is positive in the second quadrant? Choose all that apply. A B C D E F cos x sin x tan x sec x csc x cot x Unit Circle 5 Which function is positive in the fourth quadrant? Choose all that apply. Slide 23 / 207 A B C D E F cos x sin x tan x sec x csc x cot x Unit Circle Slide 24 / Which function is positive in the third quadrant? Choose all that apply. A B C D E F cos x sin x tan x sec x csc x cot x
9 Unit Circle Slide 25 / 207 Example: Given the terminal point of ( -5 / 13, -12 / 13) find sin x, cos x, and tan x. Unit Circle Slide 26 / Given the terminal point find tan x. Unit Circle Slide 27 / Given the terminal point find sin x.
10 Unit Circle Slide 28 / Given the terminal point find tan x. Unit Circle Slide 29 / Knowing sin x = Find cos x if the terminal point is in the first quadrant Unit Circle 11 Knowing sin x = Find cos x if the terminal point is in the 2 nd quadrant Slide 30 / 207
11 Slide 31 / 207 Slide 32 / 207 Graphing Return to Table of Contents Graphing Slide 33 / 207 Graphing cos, sin, & tan Graph by using values from the table. Since the values are based on a circle, values will repeat.
12 Graphing Graphing cos, sin, & tan Slide 34 / 207 Graph by using values from the table. Since the values are based on a circle, values will repeat. Graphing Graphing cos, sin, & tan Graph by using values from the table. Since the values are based on a circle, values will repeat. Slide 35 / 207 Graphing Graphing cos, sin, & tan Graph by using values from the table. Since the values are based on a circle, values will repeat. Slide 36 / 207
13 Graphing Slide 37 / 207 Parts of a trig graph cos x Amplitude x Period Slide 38 / 207 Graphing Slide 39 / 207 y= a sin(x) or y= a cos(x) In the study of transforming parent functions, we learned "a" was a vertical stretch or shrink. For trig functions it is called the amplitude.
14 Graphing Slide 40 / 207 In y= cos(x), a=1 This means at any time, y= cos (x) is at most 1 away from the axis it is oscillating about. Find the amplitude: y= 3 sin(x) y= 2 cos(x) y= -4 sin(x) Graphing Slide 41 / What is the amplitude of y = 3cosx? Graphing Slide 42 / What is the amplitude of y = 0.25cosx?
15 Graphing Slide 43 / What is the amplitude of y = -sinx? Slide 44 / 207 Graphing Slide 45 / 207 y= sin b(x) or y= cos b(x) In the study of transforming parent functions, we learned "b" was a horizontal stretch or shrink. y= cos x has b=1. Therefore cos x can make one complete cycle is 2#. For trig functions it is called the period.
16 Graphing Slide 46 / 207 y = cos x completes 1 "cycle" in 2#. So the period is 2π. y = cos 2x completes 2 "cycles" in 2# or 1 "cycle" in #. The period is # y = cos 0.5x completes 1 / 2 a cycle in 2#. The period is 4#. Graphing Slide 47 / 207 The period for y= cos bx or y= sin bx is Graphing Slide 48 / What is the period of A B C D
17 Graphing Slide 49 / What is the period of A B C D Graphing Slide 50 / What is the period of A B C D Slide 51 / 207
18 Graphing Slide 52 / 207 y= sin (x+c) or y= cos (x+c) In the study of transforming parent functions, we learned "c" was a horizontal shift y= cos (x+# ) has c = π. The graph of y= cos (x+π) is the graph of y=cos(x) shifted to the left #. For trig functions it is called the phase shift. Slide 53 / 207 Slide 54 / 207
19 Slide 55 / 207 Slide 56 / 207 Slide 57 / 207
20 Slide 58 / 207 Graphing Slide 59 / 207 y= sin (x) + d or y= cos (x) + d In the study of transforming parent functions, we learned "d" was a vertical shift Graphing 23 What is the vertical shift in Slide 60 / 207
21 Graphing Slide 61 / What is the vertical shift in Graphing Slide 62 / What is the vertical shift in Slide 63 / 207
22 Slide 64 / 207 Slide 65 / 207 Slide 66 / 207
23 Slide 67 / 207 Slide 68 / 207 Graphing Slide 69 / What is the amplitude of this cosine graph?
24 Graphing Slide 70 / What is the period of this cosine graph? (use 3.14 for pi) Graphing Slide 71 / What is the phase shift of this cosine graph? Graphing Slide 72 / What is the vertical shift of this cosine graph?
25 Graphing Slide 73 / Which of the following of the following are equations for the graph? A B C D Slide 74 / 207 Law of Sines Return to Table of Contents Slide 75 / 207
26 Law of Sines Slide 76 / 207 When to use Law of Sines (Recall triangle congruence statements) ASA AAS SAS (use Law of Cosines) SSS (use Law of Cosines) SSA (use Law of Sines- but be cautious!) Slide 77 / 207 Slide 78 / 207
27 Law of Sines Slide 79 / 207 Example: Teddy is driving toward the Old Man of the Mountain, the angle of elevation is 10 degrees, he drives another mile and the angle of elevation is 30 degrees. How tall is the mountain? y x Slide 80 / 207 Slide 81 / 207
28 Slide 82 / 207 Law of Sines with SSA. SSA information will lead to 0, 1,or 2 possible solutions. The one solution answer comes from when the bigger given side is opposite the given angle. The 2 solution and no solution come from when sin -1 is used in the problem and the answer and its supplement are evaluated, sometimes both will work, sometimes one will work,and sometimes neither will work. Law of Sines Slide 83 / 207 Example B solve triangle ABC A C Law of Sines Slide 84 / 207 Example B solve triangle ABC 7 5 A 40 C
29 Law of Sines Slide 85 / 207 Solution 1 B A C Solution 2 B 7 5 A 40 C Law of Sines Slide 86 / 207 Example solve triangle ABC B A C Slide 87 / 207
30 Law of Sines Slide 88 / How many triangles meet the following conditions? Law of Sines Slide 89 / How many triangles meet the following conditions? Slide 90 / 207 Law of Cosines Return to Table of Contents
31 Slide 91 / 207 Law of Cosines Slide 92 / 207 When we began to study Law of Sines, we looked at this table: When to use Law of Sines (Recall triangle congruence statements) ASA AAS SAS (use Law of Cosines) SSS (use Law of Cosines) SSA (use Law of Sines- but be cautious!) Its now time to look at SAS and SSS triangles. Slide 93 / 207
32 Slide 94 / 207 Slide 95 / 207 Slide 96 / 207
33 Law of Cosines Slide 97 / 207 Example: Joe went camping. Sitting at his camp site he noticed it was 3 miles to one end of the lake and 4 miles to the other end. He determined that the angle between these two line of sites is 105 degrees. How far is it across the lake? x Slide 98 / 207 Slide 99 / 207
34 Slide 100 / 207 Slide 101 / 207 Identities Return to Table of Contents Slide 102 / 207 Trigonometry Identities are useful for simplifying expressions and proving other identities.
35 Slide 103 / 207 Pythagorean Identities Return to Table of Contents Pythagorean Identities Slide 104 / 207 Trigonometric Ratios Pythagorean Identities Slide 105 / 207 Pythagorean Identities
36 Pythagorean Identities Slide 106 / 207 Simplify: Pythagorean Identities Slide 107 / 207 Simplify: Pythagorean Identities Slide 108 / 207 Simplify:
37 Pythagorean Identities Slide 109 / 207 Prove: Slide 110 / 207 Pythagorean Identities Slide 111 / 207 Prove:
38 Slide 112 / 207 Pythagorean Identities Slide 113 / The following expression can be simplified to which choice? A B C D Pythagorean Identities Slide 114 / The following expression can be simplified to which choice? A B C D
39 Pythagorean Identities Slide 115 / The following expression can be simplified to which choice? A B C D Slide 116 / 207 Angle Sum/Difference Return to Table of Contents Angle Sum/Difference Slide 117 / 207 Angle Sum/Difference Identities are used to convert angles we aren't familiar with to ones we are (ie. multiples of 30, 45, 60, & 90).
40 Angle Sum/Difference Slide 118 / 207 Sum/ Difference Identities Angle Sum/Difference Slide 119 / 207 Find the exact value of Slide 120 / 207
41 Angle Sum/Difference Slide 121 / 207 Find the exact value of Angle Sum/Difference Slide 122 / 207 Find the exact value of Angle Sum/Difference Slide 123 / 207 Prove:
42 Angle Sum/Difference Slide 124 / 207 Prove: Angle Sum/Difference Slide 125 / Which choice is another way to write the given expression? A B C D Angle Sum/Difference Slide 126 / Which choice is the exact value of the given expression? A B C D
43 Slide 127 / 207 Double Angle Return to Table of Contents Double Angle Slide 128 / 207 Double-Angle Identities Slide 129 / 207
44 Double Angle Slide 130 / 207 Write cos3x in terms of cosx Double Angle Slide 131 / Which of the following choices is equivalent to the given expression? A B C D Slide 132 / 207
45 Double Angle Slide 133 / Which of the following choices is equivalent to the given expression? A B C D Slide 134 / 207 Half Angle Return to Table of Contents Slide 135 / 207
46 Half Angle Slide 136 / 207 Find the exact value of cos15 using Half-Angle Identity Half Angle Slide 137 / 207 Find the exact value of tan 22.5 Half Angle 51 Find the exact value of Slide 138 / 207 A B C D
47 Half Angle Slide 139 / Find the exact value of A B C D Half Angle Slide 140 / 207 Find cos( u / 2) if sin u= - 3 / 7 and u is in the third quadrant Pythagorean Identity but Why Negative? Slide 141 / 207
48 Half Angle Slide 142 / Find if and u is in the 4th quadrant? A B C D Slide 143 / 207 Power Reducing Identities Return to Table of Contents Power Reducing Identities Slide 144 / 207 Power Reducing Identities
49 Power Reducing Identities Slide 145 / 207 Reduce sin 4 x to an expression in terms of first power cosines. Power Reducing Identities Slide 146 / 207 Reduce cos 4 x to an expression in terms of first power cosines. Power Reducing Identities Slide 147 / Which of the following choices is equivalent to the given expression? A B C D
50 Power Reducing Identities Slide 148 / Which of the following choices is equivalent to the given expression? A B C D Power Reducing Identities Slide 149 / Which of the following choices is equivalent to the given expression? A B C D Slide 150 / 207 Sum to Product Return to Table of Contents
51 Sum to Product Slide 151 / 207 Sum to Product Sum to Product Slide 152 / 207 Write cos 11x + cos 9x as a product Sum to Product Slide 153 / 207 Write sin 8x - sin 4x as a product
52 Sum to Product Slide 154 / 207 Find the exact value of cos 5π / 12 + cos π / 12 Sum to Product Slide 155 / 207 Prove Sum to Product Slide 156 / 207 Prove:
53 Sum to Product Slide 157 / Which of the following is equivalent to the given expression? A B C D Sum to Product Slide 158 / Which of the following is equivalent to the given expression? A B C D Sum to Product Slide 159 / Which of the following is not equivalent to the given expression? A B C D
54 Slide 160 / 207 Product to Sum Return to Table of Contents Slide 161 / 207 Product to Sum Slide 162 / 207 Rewrite as a sum of trig functions.
55 Product to Sum Slide 163 / 207 Rewrite as a sum of trig functions. Product to Sum Slide 164 / Which choice is equivalent to the expression given? A B C D Slide 165 / 207
56 Slide 166 / 207 Inverse Trig Functions Return to Table of Contents Slide 167 / 207 Slide 168 / 207
57 Inverse Trig Functions Slide 169 / 207 Inverse Trig Functions Since the cosine function does not pass the horizontal line test, we need to restrict its domain so that cos -1 is a function. cos x: Domain[0, # ] Range[-1, 1] cos -1 x: Domain[-1, 1] Range[0, π] Remember to find an inverse, switch x and y. Inverse Trig Functions Slide 170 / 207 y=cos -1 x # # /2-1 1 Inverse Trig Functions Slide 171 / 207 Inverse Trig Functions Since the sine function does not pass the horizontal line test, we need to restrict its domain so that sin -1 is a function. sin x: Domain Range[-1, 1] sin -1 x: Domain[-1, 1] Range
58 Inverse Trig Functions Slide 172 / 207 y=sin -1 x -1 1 Inverse Trig Functions Slide 173 / 207 Inverse Trig Functions Since the tangent function does not pass the horizontal line test, we need to restrict its domain so that tan -1 is a function. tan x: Domain Range tan -1 x: Domain Range Inverse Trig Functions Slide 174 / 207 y=tan -1 x
59 Inverse Trig Functions Slide 175 / 207 Secant Inverse Trig Functions Slide 176 / 207 y=sec -1 x -1 1 sec -1 x : Domain: (-#,-1] [1, # ) Range: [0, # /2) [#, 3# /2) Inverse Trig Functions Slide 177 / 207 Cosecant
60 Inverse Trig Functions Slide 178 / 207 Cosecant -1 1 sec -1 x : Domain: (-#,-1] [1, # ) Range: (0, # /2] (#, 3# /2] Inverse Trig Functions Slide 179 / 207 Cotangent Inverse Trig Functions Slide 180 / 207 Cotangent -1 1 cot -1 x: Domain: Reals Range: (0, # )
61 Inverse Trig Functions Slide 181 / 207 Restrictions Inverse Trig Functions Slide 182 / 207 Example: Evaluate the following expression. Inverse Trig Functions Slide 183 / 207 Example: Evaluate the following expression.
62 Inverse Trig Functions Slide 184 / 207 Example: Evaluate the following expressions. Inverse Trig Functions Slide 185 / Evaluate the following expression: A B C D Inverse Trig Functions Slide 186 / Evaluate the following expression: A B C D
63 Inverse Trig Functions Slide 187 / Evaluate the following expression: A B C D Inverse Trig Functions Slide 188 / 207 Example: Evaluate the following expressions. Inverse Trig Functions Slide 189 / 207 Example: Evaluate the following expressions.
64 Slide 190 / 207 Slide 191 / 207 Slide 192 / 207
65 Slide 193 / 207 Trig Equations Return to Table of Contents Trig Equations Slide 194 / 207 To solve a trigonometry equation, apply the rules of algebra to isolate the trig function(s). Examples: Solve. Trig Equations Slide 195 / 207 To solve a trigonometry equation, apply the rules of algebra to isolate the trig function(s). Examples: Solve.
66 Trig Equations Slide 196 / 207 Examples: Solve. Trig Equations Slide 197 / 207 Examples: Solve. Trig Equations Slide 198 / Find an apporoximate value of x on [0, ) that satisfies the following equation:
67 Slide 199 / 207 Trig Equations Slide 200 / 207 Examples: Solve. Trig Equations Slide 201 / 207 Examples: Solve.
68 Trig Equations Slide 202 / 207 Examples: Solve. Trig Equations Slide 203 / 207 Examples: Solve. Trig Equations Slide 204 / 207 Examples: Solve.
69 Trig Equations Slide 205 / 207 Examples: Solve. Trig Equations Slide 206 / Find an apporoximate value of x on [0, ) that satisfies the following equation: Slide 207 / 207
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