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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


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Pre-Calc

Trigonometry

www.njctl.org 2015-03-24

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Table of Contents

Unit Circle Graphing Law of Sines Law of Cosines Pythagorean Identities Angle Sum/Difference Double Angle Half Angle Sum to Product Inverse Trig Functions Trig Equations Product to Sum Power Reducing

click on the topic to go to that section

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Unit Circle

Return to Table of Contents

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Unit Circle

Goals and Objectives

Students will understand how to use the Unit Circle to find angles and determine their trigonometric value.

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Why do we need this?

The Unit Circle is a tool that allows us to determine the location of any angle. Unit Circle

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Special Right Triangles

Unit Circle

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Example 1: Find a Example 2: Find b & c 6 a 4 b c

Unit Circle

Teacher Teacher

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Example 3: Find d Example 4: Find e 8 d e 9

Unit Circle

Teacher Teacher

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Example 5: Find f Example 6: Find g & h 1 f 1 h g

Unit Circle

Teacher Teacher

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30o 45o 60o 30o 45o 60o 30o 45o 60o 30o 45o 60o

Unit Circle

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Unit Circle

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SLIDE 5

Unit Circle

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Unit Circle

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Unit Circle

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4 Which function is positive in the second quadrant? Choose all that apply.

A

cos x

B

sin x

C

tan x

D

sec x

E

csc x

F

cot x

Unit Circle

Teacher Teacher

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5 Which function is positive in the fourth quadrant? Choose all that apply.

A

cos x

B

sin x

C

tan x

D

sec x

E

csc x

F

cot x

Unit Circle

Teacher Teacher

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6 Which function is positive in the third quadrant? Choose all that apply.

A

cos x

B

sin x

C

tan x

D

sec x

E

csc x

F

cot x

Unit Circle

Teacher Teacher

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SLIDE 9

Teacher Teacher

Example: Given the terminal point of (

  • 5/13,-12/13)

find sin x, cos x, and tan x.

Unit Circle

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7 Given the terminal point find tan x.

Unit Circle

Teacher Teacher

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8 Given the terminal point find sin x.

Unit Circle

Teacher Teacher

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9 Given the terminal point find tan x.

Unit Circle

Teacher Teacher

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10 Knowing sin x = Find cos x if the terminal point is in the first quadrant

Unit Circle

Teacher Teacher

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11 Knowing sin x = Find cos x if the terminal point is in the 2

nd quadrant

Unit Circle

Teacher Teacher

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Graphing

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Graphing cos, sin, & tan Graph by using values from the table. Since the values are based on a circle, values will repeat.

Graphing

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Graphing cos, sin, & tan Graph by using values from the table. Since the values are based on a circle, values will repeat.

Graphing

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Graphing cos, sin, & tan Graph by using values from the table. Since the values are based on a circle, values will repeat.

Graphing

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Graphing cos, sin, & tan Graph by using values from the table. Since the values are based on a circle, values will repeat.

Graphing

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Parts of a trig graph

x cos x

Amplitude Period

Graphing

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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.

Graphing

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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

Teacher Teacher

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13 What is the amplitude of y = 3cosx ?

Graphing

Teacher Teacher

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14 What is the amplitude of y = 0.25cosx ?

Graphing

Teacher Teacher

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15 What is the amplitude of y = -sinx ?

Graphing

Teacher Teacher

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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.

Graphing

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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

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The period for y= cos bx or y= sin bx is

Graphing

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16 What is the period of

A B C D

Graphing

Teacher Teacher

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17 What is the period of

A B C D

Graphing

Teacher Teacher

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18 What is the period of

A B C D

Graphing

Teacher Teacher

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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.

Graphing

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y= sin (x) + d or y= cos (x) + d In the study of transforming parent functions, we learned "d" was a vertical shift

Graphing

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23 What is the vertical shift in

Graphing

Teacher Teacher

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24 What is the vertical shift in

Graphing

Teacher Teacher

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25 What is the vertical shift in

Graphing

Teacher Teacher

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30 What is the amplitude of this cosine graph?

Graphing

Teacher Teacher

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31 What is the period of this cosine graph? (use 3.14 for pi)

Graphing

Teacher Teacher

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32 What is the phase shift of this cosine graph?

Graphing

Teacher Teacher

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33 What is the vertical shift of this cosine graph?

Graphing

Teacher Teacher

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34 Which of the following of the following are equations for the graph?

A B C D

Graphing

Teacher Teacher

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Law of Sines

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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!)

Law of Sines

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Teacher Teacher

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? 30 10 5280 x y

Law of Sines

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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

  • pposite 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.

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Teacher Teacher

Example solve triangle ABC A B C 40 5 7

Law of Sines

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Teacher Teacher

Example solve triangle ABC A B C 40 7 5

Law of Sines

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Teacher Teacher

A B C 40 7 5 64.1 Solution 1 115.9 A B C 40 7 5 Solution 2

Law of Sines

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Teacher Teacher

Example solve triangle ABC A B C 50 14 7

Law of Sines

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38 How many triangles meet the following conditions?

Law of Sines

Teacher Teacher

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39 How many triangles meet the following conditions?

Law of Sines

Teacher Teacher

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Law of Cosines

Return to Table of Contents

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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!) When we began to study Law of Sines, we looked at this table: Its now time to look at SAS and SSS triangles.

Law of Cosines

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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?

3 4 105 x

Law of Cosines

Teacher Teacher

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Identities

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Trigonometry Identities are useful for simplifying expressions and proving

  • ther identities.

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Pythagorean Identities

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Trigonometric Ratios

Pythagorean Identities

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Pythagorean Identities

Pythagorean Identities

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Simplify:

Pythagorean Identities

Teacher Teacher

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Simplify:

Pythagorean Identities

Teacher Teacher

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Simplify:

Pythagorean Identities

Teacher Teacher

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Prove:

Pythagorean Identities

Teacher Teacher

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Prove:

Pythagorean Identities

Teacher Teacher

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43 The following expression can be simplified to which choice?

A B C D

Pythagorean Identities

Teacher Teacher

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44 The following expression can be simplified to which choice?

A B C D

Pythagorean Identities

Teacher Teacher

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45 The following expression can be simplified to which choice?

A B C D

Pythagorean Identities

Teacher Teacher

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Angle Sum/Difference

Return to Table of Contents

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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).

Angle Sum/Difference

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Sum/ Difference Identities

Angle Sum/Difference

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Find the exact value of

Angle Sum/Difference

Teacher Teacher

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Find the exact value of

Angle Sum/Difference

Teacher Teacher

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Find the exact value of

Angle Sum/Difference

Teacher Teacher

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Prove:

Angle Sum/Difference

Teacher Teacher

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Prove:

Angle Sum/Difference

Teacher Teacher

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46 Which choice is another way to write the given expression?

A B C D

Angle Sum/Difference

Teacher Teacher

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47 Which choice is the exact value of the given expression?

A B C D

Angle Sum/Difference

Teacher Teacher

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Double Angle

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Double-Angle Identities

Double Angle

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Write cos3x in terms of cosx

Double Angle

Teacher Teacher

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48 Which of the following choices is equivalent to the given expression?

A B C D

Double Angle

Teacher Teacher

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50 Which of the following choices is equivalent to the given expression?

A B C D

Double Angle

Teacher Teacher

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Half Angle

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Find the exact value of cos15 using Half-Angle Identity

Half Angle

Teacher Teacher

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Find the exact value of tan 22.5

Half Angle

Teacher Teacher

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51 Find the exact value of

A B C D

Half Angle

Teacher Teacher

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52 Find the exact value of

A B C D

Half Angle

Teacher Teacher

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Find cos(u/2) if sin u= - 3/7 and u is in the third quadrant

Pythagorean Identity but Why Negative?

Half Angle

Teacher Teacher

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54 Find if and u is in the 4th quadrant?

A B C D

Half Angle

Teacher Teacher

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Power Reducing Identities

Return to Table of Contents

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Power Reducing Identities

Power Reducing Identities

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Reduce sin4x to an expression in terms of first power cosines.

Power Reducing Identities

Teacher Teacher

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Reduce cos4x to an expression in terms of first power cosines.

Power Reducing Identities

Teacher Teacher

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55 Which of the following choices is equivalent to the given expression?

A B C D

Power Reducing Identities

Teacher Teacher

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56 Which of the following choices is equivalent to the given expression?

A B C D

Power Reducing Identities

Teacher Teacher

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57 Which of the following choices is equivalent to the given expression?

A B C D

Power Reducing Identities

Teacher Teacher

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Sum to Product

Return to Table of Contents

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Sum to Product

Sum to Product

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Write cos 11x + cos 9x as a product

Sum to Product

Teacher Teacher

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Write sin 8x - sin 4x as a product

Sum to Product

Teacher Teacher

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Find the exact value of cos

5π/ 12 + cos π/12

Sum to Product

Teacher Teacher

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Prove

Sum to Product

Teacher Teacher

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Prove:

Sum to Product

Teacher Teacher

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58 Which of the following is equivalent to the given expression?

A B C D

Sum to Product

Teacher Teacher

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59 Which of the following is equivalent to the given expression?

A B C D

Sum to Product

Teacher Teacher

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60 Which of the following is not equivalent to the given expression?

A B C D

Sum to Product

Teacher Teacher

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Product to Sum

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Rewrite as a sum of trig functions.

Product to Sum

Teacher Teacher

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Rewrite as a sum of trig functions.

Product to Sum

Teacher Teacher

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61 Which choice is equivalent to the expression given?

A B C D

Product to Sum

Teacher Teacher

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Inverse Trig Functions

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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

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1

  • 1

# # /2

y=cos-1x

Inverse Trig Functions

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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

Inverse Trig Functions

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y=sin-1x

1

  • 1

Inverse Trig Functions

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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

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y=tan

  • 1x

Inverse Trig Functions

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Secant

Inverse Trig Functions

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y=sec -1 x 1

  • 1

sec-1x : Domain: (-# ,-1] ∪ [1 , # ) Range: [0, # /2) ∪ [# , 3# /2)

Inverse Trig Functions

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Cosecant

Inverse Trig Functions

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1

  • 1

Cosecant sec-1x : Domain: (-# ,-1] ∪ [1 , # ) Range: (0, # /2] ∪ (# , 3# /2]

Inverse Trig Functions

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Cotangent

Inverse Trig Functions

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Cotangent 1

  • 1

cot-1 x: Domain: Reals Range: (0 , # )

Inverse Trig Functions

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Restrictions

Inverse Trig Functions

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Example: Evaluate the following expression.

Inverse Trig Functions

Teacher Teacher

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Example: Evaluate the following expression.

Inverse Trig Functions

Teacher Teacher

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Example: Evaluate the following expressions.

Inverse Trig Functions

Teacher Teacher

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63 Evaluate the following expression:

A B C D

Inverse Trig Functions

Teacher Teacher

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64 Evaluate the following expression:

A B C D

Inverse Trig Functions

Teacher Teacher

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65 Evaluate the following expression:

A B C D

Inverse Trig Functions

Teacher Teacher

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Example: Evaluate the following expressions.

Inverse Trig Functions

Teacher Teacher

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Example: Evaluate the following expressions.

Inverse Trig Functions

Teacher Teacher

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Trig Equations

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To solve a trigonometry equation, apply the rules of algebra to isolate the trig function(s).

Examples: Solve.

Trig Equations

Teacher Teacher

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To solve a trigonometry equation, apply the rules of algebra to isolate the trig function(s).

Examples: Solve.

Trig Equations

Teacher Teacher

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SLIDE 66

Examples: Solve.

Trig Equations

Teacher Teacher

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Examples: Solve.

Trig Equations

Teacher Teacher

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69 Find an apporoximate value of x on [0, ) that satisfies the following equation:

Trig Equations

Teacher Teacher

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Examples: Solve.

Trig Equations

Teacher Teacher

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Examples: Solve.

Trig Equations

Teacher Teacher

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Examples: Solve.

Trig Equations

Teacher Teacher

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Examples: Solve.

Trig Equations

Teacher Teacher

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Examples: Solve.

Trig Equations

Teacher Teacher

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Examples: Solve.

Trig Equations

Teacher Teacher

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71 Find an apporoximate value of x on [0, ) that satisfies the following equation:

Trig Equations

Teacher Teacher

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