Pre Calc. Trigonometry.
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1 1
2 Pre Calc Trigonometry
3 Table of Contents 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 3
4 Unit Circle Return to Table of Contents 4
5 Unit Circle Goals and Objectives Students will understand how to use the Unit Circle to find angles and determine their trigonometric value. 5
6 Unit Circle Why do we need this? The Unit Circle is a tool that allows us to determine the location of any angle. 6
7 Unit Circle Special Right Triangles 7
8 Unit Circle Example 1: Find a Example 2: Find b & c 6 a 4 c b 8
9 Unit Circle Example 3: Find d 8 d Example 4: Find e 9 e 9
10 Unit Circle Example 5: Find f Example 6: Find g & h f 1 g h 1 10
11 Unit Circle 30 o 45 o 60 o 60 o 45 o 30 o 30 o 30 o 45 o 60 o 60 o 45 o 11
12 Unit Circle 12
13 Unit Circle 13
14 Unit Circle 14
15 Unit Circle 15
16 Unit Circle Unit Circle 16
17 Unit Circle 17
18 Unit Circle 1 18
19 Unit Circle 1 A B C D E F G H I J 19
20 Unit Circle 2 A B C D E F G H I J 20
21 Unit Circle 3 A B C D E F G H I J 21
22 Unit Circle 4 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 22
23 Unit Circle 5 Which function is positive in the fourth quadrant? Choose all that apply. A B C D E F cos x sin x tan x sec x csc x cot x 23
24 Unit Circle 6 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 24
25 Unit Circle Example: Given the terminal point of ( 5 / 13, 12 / 13 ) find sin x, cos x, and tan x. 25
26 Unit Circle 7 Given the terminal point find tan x. 26
27 Unit Circle 8 Given the terminal point find sin x. 27
28 Unit Circle 9 Given the terminal point find tan x. 28
29 Unit Circle 10 Knowing sin x = Find cos x if the terminal point is in the first quadrant 29
30 Unit Circle 11 Knowing sin x = Find cos x if the terminal point is in the 2 nd quadrant 30
31 Unit Circle 12 Knowing tan x = Find cos x if the terminal point is in the 2 nd quadrant 31
32 Graphing Return to Table of Contents 32
33 Graphing Graphing cos, sin, & tan Graphby using values from the table. Since the values are based on a circle, values will repeat. 33
34 Graphing Graphing cos, sin, & tan Graphby using values from the table. Since the values are based on a circle, values will repeat. 34
35 Graphing Graphing cos, sin, & tan Graphby using values from the table. Since the values are based on a circle, values will repeat. 35
36 Graphing Graphing cos, sin, & tan Graphby using values from the table. Since the values are based on a circle, values will repeat. 36
37 Graphing Parts of a trig graph cos x Amplitude x Period 37
38 Graphing Recall: To combine transformation follow order of operations: Horizontal stretch of b, followed by horizontal slide of c, followed by a vertical stretch of a, and followed by a vertical shift of d. 38
39 Graphing 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. 39
40 Graphing 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) 40
41 Graphing 13 What is the amplitude of y = 3cosx? 41
42 Graphing 14 What is the amplitude of y = 0.25cosx? 42
43 Graphing 15 What is the amplitude of y = sinx? 43
44 Graphing Recall: To combine transformation follow order of operations: Horizontal stretch of b, followed by horizontal slide of c, followed by a vertical stretch of a, and followed by a vertical shift of d. 44
45 Graphing 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 2π. For trig functions it is called the period. 45
46 Graphing y = cos x completes 1 "cycle" in 2π. So the period is 2π. y = cos 2x completes 2 "cycles" in 2π or 1 "cycle" in π. The period is π y = cos 0.5x completes 1 / 2 a cycle in 2π. The period is 4π. 46
47 Graphing The period for y= cos bx or y= sin bx is 47
48 Graphing 16 What is the period of A B C D 48
49 Graphing 17 What is the period of A B C D 49
50 Graphing 18 What is the period of A B C D 50
51 Graphing Recall: To combine transformation follow order of operations: Horizontal stretch of b, followed by horizontal slide of c, followed by a vertical stretch of a, and followed by a vertical shift of d. 51
52 Graphing 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+π) has c = π. The graph of y= cos (x+π) is the graph of y=cos(x) shifted to the left π. For trig functions it is called the phase shift. 52
53 Graphing 19 What is the phase shift for the following function use the appropriate sign to indicate direction. 53
54 Graphing 20 What is the phase shift for the following function use the appropriate sign to indicate direction. 54
55 Graphing In our study of transforming parent functions, recall b is multiplied to the x and the phase shift. Before a phase shift can be determined, b has to be factored. Phase shift is π / 2 55
56 Graphing 21 What is the phase shift for the following function use the appropriate sign to indicate direction. 56
57 Graphing 22 What is the phase shift for the following function use the appropriate sign to indicate direction. 57
58 Graphing Recall: To combine transformation follow order of operations: Horizontal stretch of b, followed by horizontal slide of c, followed by a vertical stretch of a, and followed by a vertical shift of d. 58
59 Graphing y= sin (x) + d or y= cos (x) + d In the study of transforming parent functions, we learned "d" was a vertical shift 59
60 Graphing 23 What is the vertical shift in 60
61 Graphing 24 What is the vertical shift in 61
62 Graphing 25 What is the vertical shift in 62
63 Graphing Recall: To combine transformation follow order of operations: Horizontal stretch of b, followed by horizontal slide of c, followed by a vertical stretch of a, and followed by a vertical shift of d. 63
64 Graphing Find the amplitude, period, phase shift and vertical slide of the following. 64
65 Graphing 26 What is the amplitude of 65
66 Graphing 27 What is the period of 66
67 Graphing 28 What is the phase shift of 67
68 Graphing 29 What is the vertical shift of 68
69 Graphing 30 What is the amplitude of this cosine graph? 69
70 Graphing 31 What is the period of this cosine graph? (use 3.14 for pi) 70
71 Graphing 32 What is the phase shift of this cosine graph? 71
72 Graphing 33 What is the vertical shift of this cosine graph? 72
73 Graphing 34 Which of the following of the following are equations for the graph? A B C D 73
74 Law of Sines Return to Table of Contents 74
75 Law of Sines Law of Sines b A c C a B 75
76 Law of Sines 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!) 76
77 Law of Sines Law of Sines with ASA Example: Draw an approximate diagram: B 60 12, solve triangle ABC. A 40 C 77
78 Law of Sines Law of Sines with SAA Example: B 97,solve triangle ABC. A 25 8 C 78
79 Law of Sines 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 79
80 Law of Sines 35 Find b given 80
81 Law of Sines 36 Find b given 81
82 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. 82
83 Law of Sines Example B solve triangle ABC A C 83
84 Law of Sines Example B solve triangle ABC A C 84
85 Law of Sines Solution 1 B Solution 2 B A C A C 5 85
86 Law of Sines Example B solve triangle ABC 14 7 A 50 C 86
87 Law of Sines 37 How many triangles meet the following conditions? 87
88 Law of Sines 38 How many triangles meet the following conditions? 88
89 Law of Sines 39 How many triangles meet the following conditions? 89
90 Law of Cosines Return to Table of Contents 90
91 Law of Cosines Law of Cosines B c a A b C 91
92 Law of Cosines 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. 92
93 Law of Cosines SSS 6 B 7 A C 8 In SSS triangles, any of the angles can be found first. So let' start with Notice the cos A is on the far end of the equation an on the opposite side of the equation is the side opposite A. 93
94 Law of Cosines SSS solution 94
95 Law of Cosines A SAS 7 B C In an SAS triangle, the side opposite the included angle is found first. Notice the cos 100 is on the far end of the equation an on the opposite side of the equation is the side opposite B. 95
96 Law of Cosines SAS solution 96
97 Law of Cosines 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 97
98 Law of Cosines 40 find a 98
99 Law of Cosines 41 find A 99
100 Law of Cosines 42 find B 100
101 Identities Return to Table of Contents 101
102 Trigonometry Identities are useful for simplifying expressions and proving other identities. 102
103 Pythagorean Identities Return to Table of Contents 103
104 Pythagorean Identities Trigonometric Ratios 104
105 Pythagorean Identities Pythagorean Identities 105
106 Pythagorean Identities Simplify: 106
107 Pythagorean Identities Simplify: 107
108 Pythagorean Identities Simplify: 108
109 Pythagorean Identities Prove: 109
110 Pythagorean Identities Prove: 110
111 Pythagorean Identities Prove: 111
112 Pythagorean Identities Prove: 112
113 Pythagorean Identities 43 The following expression can be simplified to which choice? A B C D 113
114 Pythagorean Identities 44 The following expression can be simplified to which choice? A B C D 114
115 Pythagorean Identities 45 The following expression can be simplified to which choice? A B C D 115
116 Angle Sum/Difference Return to Table of Contents 116
117 Angle Sum/Difference 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). 117
118 Angle Sum/Difference Sum/ Difference Identities 118
119 Angle Sum/Difference Find the exact value of 119
120 Angle Sum/Difference Find the exact value of 120
121 Angle Sum/Difference Find the exact value of 121
122 Angle Sum/Difference Find the exact value of 122
123 Angle Sum/Difference Prove: 123
124 Angle Sum/Difference Prove: 124
125 Angle Sum/Difference 46 Which choice is another way to write the given expression? A B C D 125
126 Angle Sum/Difference 47 Which choice is the exact value of the given expression? A B C D 126
127 Double Angle Return to Table of Contents 127
128 Double Angle Double Angle Identities 128
129 Double Angle Prove: 129
130 Double Angle Write cos3x in terms of cosx 130
131 Double Angle 48 Which of the following choices is equivalent to the given expression? A B C D 131
132 Double Angle 49 Which of the following choices is equivalent to the given expression? A B C D 132
133 Double Angle 50 Which of the following choices is equivalent to the given expression? A B C D 133
134 Half Angle Return to Table of Contents 134
135 Half Angle Half Angle Identities Note: The choice of + or will depend on which quadrant x/2 is in. 135
136 Half Angle Find the exact value of cos15 using Half Angle Identity 136
137 Half Angle Find the exact value of tan
138 Half Angle 51 Find the exact value of A B C D 138
139 Half Angle 52 Find the exact value of A B C D 139
140 Half Angle Find cos( u / 2 ) if sin u= 3 / 7 and u is in the third quadrant Pythagorean Identity but Why Negative? 140
141 Half Angle 53 Find if and u is in the 2nd quadrant? A B C D 141
142 Half Angle 54 Find if and u is in the 4th quadrant? A B C D 142
143 Power Reducing Identities Return to Table of Contents 143
144 Power Reducing Identities Power Reducing Identities 144
145 Power Reducing Identities Reduce sin 4 x to an expression in terms of first power cosines. 145
146 Power Reducing Identities Reduce cos 4 x to an expression in terms of first power cosines. 146
147 Power Reducing Identities 55 Which of the following choices is equivalent to the given expression? A B C D 147
148 Power Reducing Identities 56 Which of the following choices is equivalent to the given expression? A B C D 148
149 Power Reducing Identities 57 Which of the following choices is equivalent to the given expression? A B C D 149
150 Sum to Product Return to Table of Contents 150
151 Sum to Product Sum to Product 151
152 Sum to Product Write cos 11x + cos 9x as a product 152
153 Sum to Product Write sin 8x sin 4x as a product 153
154 Sum to Product Find the exact value of cos 5π / 12 + cos π /
155 Sum to Product Prove 155
156 Sum to Product Prove: 156
157 Sum to Product 58 Which of the following is equivalent to the given expression? A B C D 157
158 Sum to Product 59 Which of the following is equivalent to the given expression? A B C D 158
159 Sum to Product 60 Which of the following is not equivalent to the given expression? A B C D 159
160 Product to Sum Return to Table of Contents 160
161 Product to Sum Product to Sum 161
162 Product to Sum Rewrite as a sum of trig functions. 162
163 Product to Sum Rewrite as a sum of trig functions. 163
164 Product to Sum 61 Which choice is equivalent to the expression given? A B C D 164
165 Product to Sum 62 Which choice is equivalent to the expression given? A B C D 165
166 Inverse Trig Functions Return to Table of Contents 166
167 Inverse Trig Functions Inverse Trig Functions Recall the definition of an inverse: The inverse trig functions follow the same rule. cos 1 is read arccosine sin 1 is read arcsine 167
168 Inverse Trig Functions Note: 168
169 Inverse Trig Functions 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, π] Range[ 1, 1] cos 1 x: Domain[ 1, 1] Range[0, π] Remember to find an inverse, switch x and y. 169
170 Inverse Trig Functions y=cos 1 x π π/
171 Inverse Trig Functions 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 171
172 Inverse Trig Functions y=sin 1 x
173 Inverse Trig Functions 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 173
174 Inverse Trig Functions y=tan 1 x 174
175 Inverse Trig Functions Secant 175
176 Inverse Trig Functions y=sec 1 x 1 1 sec 1 x : Domain: (, 1] [1, ) Range: [0, π/2) [π, 3π/2) 176
177 Inverse Trig Functions Cosecant 177
178 Inverse Trig Functions Cosecant 1 1 sec 1 x : Domain: (, 1] [1, ) Range: (0, π/2] (π, 3π/2] 178
179 Inverse Trig Functions Cotangent 179
180 Inverse Trig Functions Cotangent 1 1 cot 1 x: Domain: Reals Range: (0, π) 180
181 Inverse Trig Functions Restrictions 181
182 Inverse Trig Functions Example: Evaluate the following expression. 182
183 Inverse Trig Functions Example: Evaluate the following expression. 183
184 Inverse Trig Functions Example: Evaluate the following expressions. 184
185 Inverse Trig Functions 63 Evaluate the following expression: A B C D 185
186 Inverse Trig Functions 64 Evaluate the following expression: A B C D 186
187 Inverse Trig Functions 65 Evaluate the following expression: A B C D 187
188 Inverse Trig Functions Example: Evaluate the following expressions. 188
189 Inverse Trig Functions Example: Evaluate the following expressions. 189
190 Inverse Trig Functions 66 Evaluate the following expression: A B C D 190
191 Inverse Trig Functions 67 Evaluate the following expression: A B C D 191
192 Inverse Trig Functions 68 Evaluate the following expression: A B C D 192
193 Trig Equations Return to Table of Contents 193
194 Trig Equations To solve a trigonometry equation, apply the rules of algebra to isolate the trig function(s). Examples: Solve. 194
195 Trig Equations To solve a trigonometry equation, apply the rules of algebra to isolate the trig function(s). Examples: Solve. 195
196 Trig Equations Examples: Solve. 196
197 Trig Equations Examples: Solve. 197
198 Trig Equations 69 Find an apporoximate value of x on [0, ) that satisfies the following equation: 198
199 Trig Equations 70 Find an apporoximate value of x on [0, ) that satisfies the following equation: 199
200 Trig Equations Examples: Solve. 200
201 Trig Equations Examples: Solve. 201
202 Trig Equations Examples: Solve. 202
203 Trig Equations Examples: Solve. 203
204 Trig Equations Examples: Solve. 204
205 Trig Equations Examples: Solve. 205
206 Trig Equations 71 Find an apporoximate value of x on [0, ) that satisfies the following equation: 206
207 Trig Equations 72 Find an apporoximate value of x on [0, ) that satisfies the following equation: 207
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Slide 1 / 207 Slide 2 / 207 Pre-Calc Trigonometry 2015-03-24 www.njctl.org Slide 3 / 207 Table of Contents Unit Circle Graphing Law of Sines Law of Cosines Pythagorean Identities Angle Sum/Difference Double
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