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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 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
Slide 4 / 207 Unit Circle Return to Table of Contents
Slide 5 / 207 Unit Circle Goals and Objectives Students will understand how to use the Unit Circle to find angles and determine their trigonometric value.
Slide 6 / 207 Unit Circle Why do we need this? The Unit Circle is a tool that allows us to determine the location of any angle.
Slide 7 / 207 Unit Circle Special Right Triangles
Slide 8 / 207 Unit Circle Example 1: Find a Example 2: Find b & c Teacher 6 a 4 c b
Slide 9 / 207 Unit Circle Example 3: Find d Example 4: Find e Teacher d 8 9 e
Slide 10 / 207 Unit Circle Example 5: Find f Example 6: Find g & h Teacher f 1 g h 1
Slide 11 / 207 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
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Slide 22 / 207 Unit Circle 4 Which function is positive in the second quadrant? Choose all that apply. Teacher A B C D E F cos x sin x tan x sec x csc x cot x
Slide 23 / 207 Unit Circle 5 Which function is positive in the fourth quadrant? Choose all that apply. Teacher A B C D E F cos x sin x tan x sec x csc x cot x
Slide 24 / 207 Unit Circle 6 Which function is positive in the third quadrant? Choose all that apply. Teacher A B C D E F cos x sin x tan x sec x csc x cot x
Slide 25 / 207 Unit Circle Example: Given the terminal point of ( -5 / 13, -12 / 13 ) find sin x, cos x, and tan x. Teacher
Slide 26 / 207 Unit Circle 7 Given the terminal point find tan x. Teacher
Slide 27 / 207 Unit Circle 8 Given the terminal point find sin x. Teacher
Slide 28 / 207 Unit Circle 9 Given the terminal point find tan x. Teacher
Slide 29 / 207 Unit Circle 10 Knowing sin x = Teacher Find cos x if the terminal point is in the first quadrant
Slide 30 / 207 Unit Circle 11 Knowing sin x = Teacher Find cos x if the terminal point is in the 2 nd quadrant
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Slide 32 / 207 Graphing Return to Table of Contents
Slide 33 / 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 34 / 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 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.
Graphing Slide 36 / 207 Graphing cos, sin, & tan Graph by using values from the table. Since the values are based on a circle, values will repeat.
Slide 37 / 207 Graphing Parts of a trig graph cos x Amplitude x Period
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Slide 39 / 207 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.
Slide 40 / 207 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. Teacher Find the amplitude: y= 3 sin(x) y= 2 cos(x) y= -4 sin(x)
Slide 41 / 207 Graphing 13 What is the amplitude of y = 3cosx? Teacher
Slide 42 / 207 Graphing 14 What is the amplitude of y = 0.25cosx? Teacher
Slide 43 / 207 Graphing 15 What is the amplitude of y = -sinx? Teacher
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Slide 45 / 207 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 2#. For trig functions it is called the period.
Slide 46 / 207 Graphing 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#.
Slide 47 / 207 Graphing The period for y= cos bx or y= sin bx is
Slide 48 / 207 Graphing 16 What is the period of Teacher A B C D
Slide 49 / 207 Graphing 17 What is the period of Teacher A B C D
Slide 50 / 207 Graphing 18 What is the period of Teacher A B C D
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Slide 52 / 207 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+# ) 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.
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Slide 59 / 207 Graphing y= sin (x) + d or y= cos (x) + d In the study of transforming parent functions, we learned "d" was a vertical shift
Slide 60 / 207 Graphing 23 What is the vertical shift in Teacher
Slide 61 / 207 Graphing 24 What is the vertical shift in Teacher
Slide 62 / 207 Graphing 25 What is the vertical shift in Teacher
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Slide 69 / 207 Graphing 30 What is the amplitude of this cosine graph? Teacher
Slide 70 / 207 Graphing 31 What is the period of this cosine graph? (use 3.14 for pi) Teacher
Slide 71 / 207 Graphing 32 What is the phase shift of this cosine graph? Teacher
Slide 72 / 207 Graphing 33 What is the vertical shift of this cosine graph? Teacher
Slide 73 / 207 Graphing 34 Which of the following of the following are equations for the graph? Teacher A B C D
Slide 74 / 207 Law of Sines Return to Table of Contents
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Slide 76 / 207 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!)
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Slide 79 / 207 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? Teacher 10 5280 30 y x
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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.
Slide 83 / 207 Law of Sines Example B solve triangle ABC Teacher A 5 7 40 C
Slide 84 / 207 Law of Sines Example B solve triangle ABC Teacher A 40 7 5 C
Slide 85 / 207 Law of Sines Solution 1 B Solution 2 B Teacher A 7 5 40 64.1 C A 7 40 115.9 C 5
Slide 86 / 207 Law of Sines Example B solve triangle ABC Teacher 14 7 A 50 C
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Slide 88 / 207 Law of Sines 38 How many triangles meet the following conditions? Teacher
Slide 89 / 207 Law of Sines 39 How many triangles meet the following conditions? Teacher
Slide 90 / 207 Law of Cosines Return to Table of Contents
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Slide 92 / 207 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.
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Slide 97 / 207 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? Teacher 3 105 4 x
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Slide 101 / 207 Identities Return to Table of Contents
Slide 102 / 207 Trigonometry Identities are useful for simplifying expressions and proving other identities.
Slide 103 / 207 Pythagorean Identities Return to Table of Contents
Slide 104 / 207 Pythagorean Identities Trigonometric Ratios
Slide 105 / 207 Pythagorean Identities Pythagorean Identities
Slide 106 / 207 Pythagorean Identities Simplify: Teacher
Slide 107 / 207 Pythagorean Identities Simplify: Teacher
Slide 108 / 207 Pythagorean Identities Simplify: Teacher
Slide 109 / 207 Pythagorean Identities Prove: Teacher
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Slide 111 / 207 Pythagorean Identities Prove: Teacher
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Slide 113 / 207 Pythagorean Identities 43 The following expression can be simplified to which choice? Teacher A B C D
Slide 114 / 207 Pythagorean Identities 44 The following expression can be simplified to which choice? Teacher A B C D
Slide 115 / 207 Pythagorean Identities 45 The following expression can be simplified to which choice? Teacher A B C D
Slide 116 / 207 Angle Sum/Difference Return to Table of Contents
Slide 117 / 207 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).
Slide 118 / 207 Angle Sum/Difference Sum/ Difference Identities
Slide 119 / 207 Angle Sum/Difference Find the exact value of Teacher
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Slide 121 / 207 Angle Sum/Difference Find the exact value of Teacher
Slide 122 / 207 Angle Sum/Difference Find the exact value of Teacher
Slide 123 / 207 Angle Sum/Difference Prove: Teacher
Slide 124 / 207 Angle Sum/Difference Prove: Teacher
Slide 125 / 207 Angle Sum/Difference 46 Which choice is another way to write the given expression? Teacher A B C D
Slide 126 / 207 Angle Sum/Difference 47 Which choice is the exact value of the given expression? Teacher A B C D
Slide 127 / 207 Double Angle Return to Table of Contents
Slide 128 / 207 Double Angle Double-Angle Identities
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Slide 130 / 207 Double Angle Write cos3x in terms of cosx Teacher
Slide 131 / 207 Double Angle 48 Which of the following choices is equivalent to the given expression? Teacher A B C D
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Slide 133 / 207 Double Angle 50 Which of the following choices is equivalent to the given expression? Teacher A B C D
Slide 134 / 207 Half Angle Return to Table of Contents
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Slide 136 / 207 Half Angle Find the exact value of cos15 using Half-Angle Identity Teacher
Slide 137 / 207 Half Angle Find the exact value of tan 22.5 Teacher
Slide 138 / 207 Half Angle 51 Find the exact value of Teacher A B C D
Slide 139 / 207 Half Angle 52 Find the exact value of Teacher A B C D
Slide 140 / 207 Half Angle Find cos( u / 2 ) if sin u= - 3 / 7 and u is in the third quadrant Pythagorean Identity but Why Negative? Teacher
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Slide 142 / 207 Half Angle 54 Find if and u is in the 4th quadrant? A Teacher B C D
Slide 143 / 207 Power Reducing Identities Return to Table of Contents
Slide 144 / 207 Power Reducing Identities Power Reducing Identities
Slide 145 / 207 Power Reducing Identities Reduce sin 4 x to an expression in terms of first power cosines. Teacher
Slide 146 / 207 Power Reducing Identities Reduce cos 4 x to an expression in terms of first power cosines. Teacher
Slide 147 / 207 Power Reducing Identities 55 Which of the following choices is equivalent to the given expression? Teacher A B C D
Slide 148 / 207 Power Reducing Identities 56 Which of the following choices is equivalent to the given expression? Teacher A B C D
Slide 149 / 207 Power Reducing Identities 57 Which of the following choices is equivalent to the given expression? Teacher A B C D
Slide 150 / 207 Sum to Product Return to Table of Contents
Slide 151 / 207 Sum to Product Sum to Product
Slide 152 / 207 Sum to Product Write cos 11x + cos 9x as a product Teacher
Slide 153 / 207 Sum to Product Write sin 8x - sin 4x as a product Teacher
Slide 154 / 207 Sum to Product Find the exact value of cos 5π / 12 + cos π / 12 Teacher
Slide 155 / 207 Sum to Product Prove Teacher
Slide 156 / 207 Sum to Product Prove: Teacher
Slide 157 / 207 Sum to Product 58 Which of the following is equivalent to the given expression? Teacher A B C D
Slide 158 / 207 Sum to Product 59 Which of the following is equivalent to the given expression? Teacher A B C D
Slide 159 / 207 Sum to Product 60 Which of the following is not equivalent to the given expression? Teacher A B C D
Slide 160 / 207 Product to Sum Return to Table of Contents
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Slide 162 / 207 Product to Sum Rewrite as a sum of trig functions. Teacher
Slide 163 / 207 Product to Sum Rewrite as a sum of trig functions. Teacher
Slide 164 / 207 Product to Sum 61 Which choice is equivalent to the expression given? Teacher A B C D
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Slide 166 / 207 Inverse Trig Functions Return to Table of Contents
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Slide 169 / 207 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, # ] Range[-1, 1] cos -1 x: Domain[-1, 1] Range[0, π] Remember to find an inverse, switch x and y.
Slide 170 / 207 Inverse Trig Functions y=cos -1 x # # /2-1 1
Slide 171 / 207 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
Slide 172 / 207 Inverse Trig Functions y=sin -1 x -1 1
Slide 173 / 207 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
Slide 174 / 207 Inverse Trig Functions y=tan -1 x
Slide 175 / 207 Inverse Trig Functions Secant
Slide 176 / 207 Inverse Trig Functions y=sec -1 x -1 1 sec -1 x : Domain: (-#,-1] [1, # ) Range: [0, # /2) [#, 3# /2)
Slide 177 / 207 Inverse Trig Functions Cosecant
Slide 178 / 207 Inverse Trig Functions Cosecant -1 1 sec -1 x : Domain: (-#,-1] [1, # ) Range: (0, # /2] (#, 3# /2]
Slide 179 / 207 Inverse Trig Functions Cotangent
Slide 180 / 207 Inverse Trig Functions Cotangent -1 1 cot -1 x: Domain: Reals Range: (0, # )
Slide 181 / 207 Inverse Trig Functions Restrictions
Slide 182 / 207 Inverse Trig Functions Example: Evaluate the following expression. Teacher
Slide 183 / 207 Inverse Trig Functions Example: Evaluate the following expression. Teacher
Slide 184 / 207 Inverse Trig Functions Example: Evaluate the following expressions. Teacher
Slide 185 / 207 Inverse Trig Functions 63 Evaluate the following expression: Teacher A B C D
Slide 186 / 207 Inverse Trig Functions 64 Evaluate the following expression: Teacher A B C D
Slide 187 / 207 Inverse Trig Functions 65 Evaluate the following expression: Teacher A B C D
Slide 188 / 207 Inverse Trig Functions Example: Evaluate the following expressions. Teacher
Slide 189 / 207 Inverse Trig Functions Example: Evaluate the following expressions. Teacher
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Slide 193 / 207 Trig Equations Return to Table of Contents
Slide 194 / 207 Trig Equations To solve a trigonometry equation, apply the rules of algebra to isolate the trig function(s). Teacher Examples: Solve.
Slide 195 / 207 Trig Equations To solve a trigonometry equation, apply the rules of algebra to isolate the trig function(s). Teacher Examples: Solve.
Slide 196 / 207 Trig Equations Examples: Solve. Teacher
Slide 197 / 207 Trig Equations Examples: Solve. Teacher
Slide 198 / 207 Trig Equations 69 Find an apporoximate value of x on [0, ) that satisfies the following equation: Teacher
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Slide 200 / 207 Trig Equations Examples: Solve. Teacher
Slide 201 / 207 Trig Equations Examples: Solve. Teacher
Slide 202 / 207 Trig Equations Examples: Solve. Teacher
Slide 203 / 207 Trig Equations Examples: Solve. Teacher
Slide 204 / 207 Trig Equations Examples: Solve. Teacher
Slide 205 / 207 Trig Equations Examples: Solve. Teacher
Slide 206 / 207 Trig Equations 71 Find an apporoximate value of x on [0, ) that satisfies the following equation: Teacher
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