Slide 2 / 207 Pre-Calc -...

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Pre-Calc

Trigonometry

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2015-03-24

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Table of Contents

Unit CircleGraphingLaw of SinesLaw of CosinesPythagorean IdentitiesAngle Sum/DifferenceDouble AngleHalf Angle

Sum to Product

Inverse Trig FunctionsTrig Equations

Product to Sum

Power Reducing

click on the topic to go to that section

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Unit Circle

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Unit Circle

Goals and ObjectivesStudents 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

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Example 3: Find d Example 4: Find e

8

d

e

9

Unit Circle

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

1

f

1

hg

Unit Circle

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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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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

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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

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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

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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

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8 Given the terminal point find sin x.

Unit Circle

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9 Given the terminal point find tan x.

Unit Circle

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10 Knowing sin x =

Find cos x if the terminal point is in the first quadrant

Unit Circle

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11 Knowing sin x =

Find cos x if the terminal point is in the 2nd quadrant

Unit Circle

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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

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

Graphing

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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=1This 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

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13 What is the amplitude of y = 3cosx ?

Graphing

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14 What is the amplitude of y = 0.25cosx ?

Graphing

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15 What is the amplitude of y = -sinx ?

Graphing

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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

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A

B

C

D

Graphing

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A

B

C

D

GraphingTe

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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

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Graphing

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Graphing

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30 What is the amplitude of this cosine graph?

Graphing

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

Graphing

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32 What is the phase shift of this cosine graph?

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33 What is the vertical shift of this cosine graph?

Graphing

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

A

B

C

D

Graphing

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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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Teac

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the angle of elevation is 10 degrees, he drives another mile and the angle of elevation is 30 degrees. How tall is the mountain?

30105280

xy

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 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.

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A

B

C405 7

Law of Sines

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Example solve triangle ABC

A

B

C40

7 5

Law of Sines

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A

B

C40

7 5

64.1

Solution 1

115.9A

B

C407 5

Solution 2

Law of Sines

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Example solve triangle ABC

A

B

C50

14 7

Law of Sines

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38 How many triangles meet the following conditions?

Law of Sines

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39 How many triangles meet the following conditions?

Law of Sines

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Law of Cosines


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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 4105

x

Law of Cosines

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Identities


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Trigonometry Identities are useful for simplifying expressions and proving

other identities.

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Pythagorean Identities


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Trigonometric Ratios


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Simplify:


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Simplify:

Pythagorean IdentitiesTe

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Simplify:


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Prove:


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Prove:


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43 The following expression can be simplified to which choice?

A

B

C

D


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A

B

C

D


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A

B

C

D


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Angle Sum/Difference


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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).


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Sum/ Difference Identities


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Find the exact value of


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Angle Sum/DifferenceTe

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Prove:


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Prove:


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

A

B

C

D

Angle Sum/DifferenceTe

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

A

B

C

D


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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

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

A

B

C

D

Double Angle

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A

B

C

D

Double Angle

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Half Angle


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Find the exact value of cos15 using Half-Angle Identity

Half Angle

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Find the exact value of tan 22.5

Half Angle

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51 Find the exact value of

A

B

C

D

Half Angle

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52 Find the exact value of

A

B

C

D

Half Angle

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

Pythagorean Identity butWhy Negative?

Half Angle

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

A

B

C

D

Half Angle

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Power Reducing Identities


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


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

Power Reducing IdentitiesTe

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A

B

C

D


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A

B

C

D


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A

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D


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Sum to Product


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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

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Write sin 8x - sin 4x as a product

Sum to Product

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Find the exact value of cos 5π/ 12 + cos π/12

Sum to Product

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Prove

Sum to Product

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Prove:

Sum to Product

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

A

B

C

D

Sum to Product

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

A

B

C

D

Sum to Product

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

A

B

C

D

Sum to Product

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Product to Sum


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Rewrite as a sum of trig functions.

Product to Sum

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Rewrite as a sum of trig functions.

Product to Sum

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61 Which choice is equivalent to the expression given?

A

B

C

D

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Inverse Trig Functions


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Inverse Trig FunctionsSince 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.


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1-1

#

# /2

y=cos-1x


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Inverse Trig FunctionsSince 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


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y=sin-1x

1-1


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Inverse Trig FunctionsSince 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


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y=tan-1x


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Secant


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y=sec -1 x

1-1

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


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Cosecant


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1-1

Cosecant

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


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Cotangent


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Cotangent

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


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Restrictions


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Example: Evaluate the following expression.

Inverse Trig FunctionsTe

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Example: Evaluate the following expression.


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Example: Evaluate the following expressions.


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63 Evaluate the following expression:

A

B

C

D


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A

B

C

D


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A

B

CD


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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 EquationsTe

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

Examples: Solve.

Trig Equations

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Examples: Solve.

Trig Equations

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Examples: Solve.

Trig EquationsTe

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

Trig Equations

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Examples: Solve.

Trig EquationsTe

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Examples: Solve.

Trig Equations

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Examples: Solve.

Trig Equations

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Examples: Solve.

Trig Equations

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Examples: Solve.

Trig Equations

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Examples: Solve.

Trig Equations

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

Trig Equations

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