Pre-Calc Trigonometry 2015-03-24 www.njctl.org Slide 3 / 207 - PDF document

Slide 1 / 207 Slide 2 / 207 Pre-Calc Trigonometry 2015-03-24 www.njctl.org Slide 3 / 207 Table of Contents click on the topic to go Unit Circle to that section 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

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 Teacher Teacher Example 2: Find b & c Example 1: Find a a b 4 c 6 Slide 9 / 207 Unit Circle Teacher Teacher Example 4: Find e Example 3: Find d e 8 9 d

Slide 10 / 207 Unit Circle Teacher Teacher Example 6: Find g & h Example 5: Find f 1 1 g h f Slide 11 / 207 Unit Circle 60 o 60 o 45 o 45 o 30 o 30 o 30 o 30 o 45 o 45 o 60 o 60 o Slide 12 / 207 Unit Circle

Slide 13 / 207 Unit Circle Slide 14 / 207 Unit Circle Slide 15 / 207 Unit Circle

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Slide 22 / 207 Unit Circle 4 Which function is positive in the second quadrant? Teacher Teacher Choose all that apply. A cos x sin x B tan x C sec x D csc x E F cot x Slide 23 / 207 Unit Circle Teacher Teacher 5 Which function is positive in the fourth quadrant? Choose all that apply. cos x A sin x B tan x C sec x D E csc x cot x F Slide 24 / 207 Unit Circle Teacher Teacher 6 Which function is positive in the third quadrant? Choose all that apply. cos x A sin x B tan x C sec x D csc x E cot x F

Slide 25 / 207 Unit Circle Example: Given the terminal point of ( -5 / 13 , -12 / 13 ) find sin x, cos x, and tan x. Teacher Teacher Slide 26 / 207 Unit Circle Teacher Teacher 7 Given the terminal point find tan x. Slide 27 / 207 Unit Circle Teacher Teacher 8 Given the terminal point find sin x.

Slide 28 / 207 Unit Circle Teacher Teacher 9 Given the terminal point find tan x. Slide 29 / 207 Unit Circle Teacher Teacher 10 Knowing sin x = Find cos x if the terminal point is in the first quadrant Slide 30 / 207 Unit Circle Teacher Teacher 11 Knowing sin x = Find cos x if the terminal point is in the 2 nd quadrant

Slide 31 / 207 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. Slide 36 / 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 37 / 207 Graphing Parts of a trig graph cos x Amplitude x Period Slide 38 / 207 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 Teacher Teacher 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) Slide 41 / 207 Graphing Teacher Teacher 13 What is the amplitude of y = 3cosx ? Slide 42 / 207 Graphing Teacher Teacher 14 What is the amplitude of y = 0.25cosx ?

Slide 43 / 207 Graphing Teacher Teacher 15 What is the amplitude of y = -sinx ? Slide 44 / 207 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 Teacher Teacher 16 What is the period of A B C D

Slide 49 / 207 Graphing Teacher Teacher 17 What is the period of A B C D Slide 50 / 207 Graphing Teacher Teacher 18 What is the period of A B C D Slide 51 / 207

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. Slide 53 / 207 Slide 54 / 207

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Slide 58 / 207 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 Teacher Teacher 23 What is the vertical shift in

Slide 61 / 207 Graphing Teacher Teacher 24 What is the vertical shift in Slide 62 / 207 Graphing Teacher Teacher 25 What is the vertical shift in Slide 63 / 207

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Slide 67 / 207 Slide 68 / 207 Slide 69 / 207 Graphing Teacher Teacher 30 What is the amplitude of this cosine graph?

Slide 70 / 207 Graphing 31 What is the period of this cosine graph? (use 3.14 for pi) Teacher Teacher Slide 71 / 207 Graphing Teacher Teacher 32 What is the phase shift of this cosine graph? Slide 72 / 207 Graphing Teacher Teacher 33 What is the vertical shift of this cosine graph?

Slide 73 / 207 Graphing Teacher Teacher 34 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

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!) Slide 77 / 207 Slide 78 / 207

Slide 79 / 207 Law of Sines Example: Teddy is driving toward the Old Man of the Mountain, Teacher Teacher the angle of elevation is 10 degrees, he drives another mile and the angle of elevation is 30 degrees. How tall is the mountain? x y 10 30 5280 Slide 80 / 207 Slide 81 / 207

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 solve triangle ABC Teacher Teacher B 7 5 40 C A Slide 84 / 207 Law of Sines Teacher Teacher Example solve triangle ABC B 7 5 40 C A

Slide 85 / 207 Law of Sines Teacher Teacher Solution 2 Solution 1 B B 5 7 7 5 64.1 40 40 A C C A 115.9 Slide 86 / 207 Law of Sines Teacher Teacher solve triangle ABC Example B 14 7 50 A C Slide 87 / 207

Slide 88 / 207 Law of Sines Teacher Teacher 38 How many triangles meet the following conditions? Slide 89 / 207 Law of Sines 39 How many triangles meet the following conditions? Teacher Teacher Slide 90 / 207 Law of Cosines Return to Table of Contents

Slide 91 / 207 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. Slide 93 / 207

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Slide 97 / 207 Law of Cosines Example: Joe went camping. Sitting at his camp site he noticed Teacher Teacher 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? 105 4 3 x Slide 98 / 207 Slide 99 / 207

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.

Slide 103 / 207 Pythagorean Identities Return to Table of Contents Slide 104 / 207 Pythagorean Identities Trigonometric Ratios Slide 105 / 207 Pythagorean Identities Pythagorean Identities