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Algebra II

Trigonometric Functions

www.njctl.org

2015-12-17

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Trig Functions

· Graphing

· Radians & Degrees & Co-terminal angles

· Unit Circle

click on the topic to go to that section

· Trigonometric Identities

· Arc Length & Area of a Sector

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Radians & Degrees

and Co-Terminal Angles

Return to Table of Contents

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A few definitions:

A central angle of a circle is an angle whose vertex is the center of the circle.

An intercepted arc is the part of the circle that includes the points of intersection with the central angle and all the points in the interior of the angle.

centralangle

interceptedarc

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Radians and Degrees

One radian is the measure of a central angle that intercepts an arc whose length is equal to the radius of the circle. There are , or a little more than 6, radians in a circle.

Click on the circle for an animated view of radians.

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Converting from Degrees to Radians

360∘ = 2 radians

1∘ = = 2360 180 radians

Example: Convert 50∘ and 90∘ to radians.

50∘ ⋅ 180 = 18

radians

90∘ ⋅ 180 = 2

radians

5

There are 360 in a circle. Therefore

Use this conversion factor to covert degrees to radians.

5

radians 2=

180 90∘

radians 18=

180 50∘

Example: Convert 50∘ and 90∘ to radians.

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Converting from Radians to Degrees

2 radians = 360∘

1 radian = = 2

180

Example: Convert and to radians

360 degrees

4

4 180 ⋅ = 45∘

⋅ 180 = 180∘

Use this conversion factor to covert radians to degrees.

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Converting between Radians and Degrees

Convert degrees to radians

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Converting between Radians and Degrees

Convert radians to degrees

radians

radians

radians

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4 Convert radians to degrees:

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Angles

Initial side

Terminal side

Initial side

Terminal side

AngleAngle in standard position

An angle is formed by rotating a ray about its endpoint. The starting position is the initial side and the ending position is the terminal side.

When, on the coordinate plane, the vertex of the angle is the origin and the initial side is the positive x-axis, the angle is in standard position.

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Positive Angle - terminal side rotates in a counter-clockwise direction

Negative Angle - terminal side rotates in a clockwise direction

α = - 37∘

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Drawing angles in standard position

310∘

40∘

Each quadrant is 90∘, and 310∘ is 40∘ more than 270∘, so the terminal side is 40∘ past the negative y-axis.

500∘

500∘ is 140∘ more than 360∘, so the angle makes a complete revolution counterclockwise and then another 140∘.

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Coterminal Angles

Angles that have the same terminating side are coterminal. To find coterminal angles add or subtract multiples of 360∘ for degrees and 2 for radians.

Example: Find one positive and one negative angle that are terminal with 75∘.

75∘-285∘

435∘

75 + 360 = 435

75 - 360 = -285

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5 Which angles are coterminal with 40∘? (Select all that are correct.)

A 320B -320C 400D -400

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6 Which graph represents 425∘ ?

AB

C D

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7 Which graph represents ?

A B

CD

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8 Which angle is NOT coterminal with -55∘?

A 305∘

B 665∘

C -415∘

D -305∘

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9 Which angle is coterminal with ?

A

B

C

D

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Arc Length & Area of a Sector

Return to Table of Contents

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Arc length: s = rArea of sector: A =

Arc length and the area of a sector

rarc length s

sector

How do these formulas relate to the area and the circumference of a circle?

(Measured in radians)

Area of sector: A = Arc length: s = r

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40∘45∘

Who is getting more pie? Who is getting more of the crust at the outer edge?

Emily's slice is cut from a 9 inch pie.

Chester's slice is cut from an 8 inch pie.

(Assume both pies are the same height.)

(Try to work this out in your groups. The solution is on the next slide)

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40∘45∘

The top of Emily's piece has an area of

The top of Chester's piece has an area of

Emily's crust has a length of

Chester's crust has a length of

click click

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10 What is the top surface area of this slice of pizza from an 18-inch pie?

45∘

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11 What is the arc length of the outer edge of this slice of pizza from an 18-inch pie?

45∘

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12 If the radius of this circular saw blade is 10 inches and there are 40 teeth on the blade, how far apart are the tips of the teeth?

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13 Challenge Question: Given a dart board as shown. If a dart thrown randomly lands somewhere on the board, what is the probability that it will land on a red region?

8 inches

4 in

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Unit CircleReturn toTable of

Contents

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The circle x2 + y2 = 1, with center (0,0) and radius 1, is called the unit circle.

1

(1,0)

(0,-1)

(0,1)

(-1,0)

Quadrant I: x and y are both positive

Quadrant IV: x is positive and y is negative

Quadrant III: x and y are both negative

Quadrant II: x is negative and y is positive

The Unit Circle

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In this triangle,

sin#= = b

θa

b

(a,b)

(1,0)

(0,-1)

(-1,0)

(0,1)

b1

cos# = = aa1

so the coordinates of (a,b) are also

(cos#, sin#)

1

The unit circle allows us to extend trigonometry beyond angles of triangles to angles of all measures.

For any angle in standard position, the point where the terminal side of the angle intercepts the circle is called the terminal point.

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In this example, the terminal point is in Quadrant IV.

For any angle θ in standard position, the terminal point has coordinates (cosθ, sinθ).

0.57

0.82-55

1

If we look at the triangle, we can see that

sin(-55 ) = 0.82

cos(-55 ) = 0.57

EXCEPT that we have to take the direction into account, and so sin(-55 ) is negative because the y value is below the x-axis.

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Click the star below to go to the Khan Academy Unit Circle Manipulative try some problems:

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1

In this example, we know the angle. Using a calculator, we find that cos 44∘ ≈ .72 and sin 44∘ ≈ .69, so the coordinates of C are approximately (0.72, 0.69).

Note that 0.722 + 0.692 ≈ 1!

What are the coordinates of point C?

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The Tangent Function

Recall SOH-CAH-TOA

hypotenuse

adjacent side

oppo

site

sid

e

#

sin = # opphyp

cos = # adjhyp

tan = # oppadj

It is also true that tan = . # sin cos ##

Why? opphypadjhyp

= ⋅ = = tan opphyp adj

hyp oppadj #

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Angles in the Unit Circle

Example: Given a terminal point on the unit circle (- ).

Find the value of cos, sin and tan of the angle.

Solution: Let the angle be .

x = cos , so cos = .

y = sin , so sin = .

tan = = = = ⋅

(Shortcut: Just cross out the 41's in the complex fraction.)

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Example: Given a terminal point , find #, tan# and csc#.

To find #, use sin-1 or cos -1 :

sin -1 ( ) = #

# # 28.1∘tan# = sin#/ cos#

tan # =

csc# = 1/ sin#

csc # = Note the "hidden" Pythagorean Triple, 8, 15, 17).

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(__, - ) 513

A

θ22.3 -22.3

Example: Find the x-value of point A, θ and the tan θ. For every point on the circle,

Since x is in quadrant III, x = -1213

sin-1(- ) ≈ -22.3 , BUT θ is in quadrant III, so θ = 180 + 22.3 = 202.3 (notice how 202.3 and -22.3 have the same sine)

513

tan θ = = = sin θcos θ

512

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Example: Given the terminal point of (-5/13,-12 /13). Find sin x, cos x, and tan x.

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14 What is tan θ?

A

B

C

D

(- ,__)35

θ

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15 What is sin θ?

A

B

C

D

(- ,__)35

θ

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16 What is θ (give your answer to the nearest degree)?

(- ,__)35

θ

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17 Given the terminal point , find tan x.

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Equilateral and isosceles triangles occur frequently in geometry and trigonometry. The angles in these triangles are multiples of 30 and 45 . A calculator will give approximate values for the trig functions of these angles, but we often need to know the exact values.

Isosceles Right Triangle Equilateral Triangle

(the altitude divides the triangle into two 30-60-90 triangles)

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Special Right Triangles

(see Triangle Trig Review unit for more detail on this topic)

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1

( , )

45∘

Special Triangles and the Unit Circle

(- , )

-

1

45∘

Multiples of 45 angles have sin and cos of ± ,

depending on the quadrant.

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30o

45o

60o

30o

45o

60o

30o

45o

60o

30o

45o

60o

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Drag the degree and radian angle measures to the angles of the circle:

#4

3#4

5#4

3#2

#2# 7#

42#0

0∘

45∘

90∘

135∘

180∘

225∘

270∘

315∘

360∘

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#4

3#4

5#4

3#2

#2

#

7#4

2#0

( , )

( , )

( , )

( , )

( , )

( , )

( , )

( , )

0

1

-1

Fill in the coordinates of x and y for each point on the unit circle:

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30∘

( , )

60∘

( , )

11

Special Triangles and the Unit Circle

Angles that are multiples of 30 have sin and cos of ± and ± .

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Drag the degree and radian angle measures to the angles of the circle:

#6

2#3

5#3

3#2

#2

# 7#6

2#0 11#6

4#3

#3

5#60∘

30∘

60∘

90∘

120∘

150∘

180∘

210∘

240∘

270∘

300∘

330∘

360∘

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#6

2#3

5#3

3#2

#2

#

7#6

2#0

11#6

4#3

#3

5#6

( , )

( , )

( , )

( , )( , )( , )

( , )

( , )

( , )

( , )

Drag in the coordinates of x and y for each point on the unit circle:

( , )

( , )

0

1

-1

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Special Angles in Degrees

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Radian Values of Special Angles

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Exact Values of Special Angles

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Put it all together...

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Degrees 0∘ 30∘ 45∘ 60∘ 90∘

Radians

sin θ cos θ

tan θ

Complete the table below:

Exact values of special angles

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If we know one trig function value and the quadrant in which the angle lies, we can find the angle and the other trig values.

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opp

adj

hyp

-3

Example: If tan = , and sin < 0, find sin , cos and the value of .

Solution: Since tan is positive and sin is negative, the terminal side of must be in Quadrant III.

· Draw a right triangle in Quadrant III.

· Use the Pythagorean Theorem to find the length of the hypotenuse:

(Continued on next slide)

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opp

adj

hyp

-3Once we know the lengths for each side, we can calculate the sin, cos and the angle. Used the signed numbers to get the correct values.

sin = =

cos = =

Use any inverse trig function to find the angle. tan-1( ) ≈ 36.7 . Because the angle is in QIII, we need to add 180 + 36.7 = 216.7, so ≈ 217 .

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22 Which functions are 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

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23 Which functions are 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

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24 Which functions are 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

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Graphing Trig

Functions Return toTable of

Contents

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If you have Geogebra available on your computer, click the star below to download a geogebratube animated graph of the trig functions:

(Once the webpage opens, click on Download)

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Graphing the Sine Function, y = sin x

Graph by creating a table of values of key points. One option is to use the set of values for x that are multiples of , and the corresponding values of y or sin x.

(Remember, is just a bit more than 3.)

Since the values are based on a circle, values will repeat.

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Notice that the graph of y = sin x increases from 0 to 1, then decreases back to 0 and then to -1 and then goes back up to 0, as x increases from 0 to 2 .

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Graphing the Cosine Curve

Since the values are based on a circle, values will repeat.

Make a table of values just as we did for sin. We could use any interval, but are choosing from 0 to 2 .

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Notice that the graph of y = cos x starts at 1, decreases to -1 and then goes back up to 1 as x increases from 0 to 2 .

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Compare the graphs:

How are they similar and how are they different?

y = sin x y = cos x

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Characteristics of y = sin x and y = cos x

Domain: set of real numbers (x can be anything)

Range: -1 ≤ y ≤ 1

Amplitude: one-half the distance from the minimum of the graph to the maximum or 1.

The functions are periodic - the pattern repeats every 2 units.

amplitude = 1

period = 2

range:-1 ≤ y ≤ 1

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Predict, Explore, Confirm

1. Using your prior knowledge of transforming functions, predict what happens to the following functions:

2. Using your graphing calculator, insert the parent function into and the transformed function into . Compare the graphs.

3. Do your conclusions match your predictions?

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y = a s in x or y = a cos x

Amplitude is a positive number that represents one-half the difference between the minimum and the maximum values, or the distance from the midline to the maximum.

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Consider the graphs of y = s in x y = 2s in x y = s in x

y = s in x

y = 2s in x

y = s in x

What do you notice about these graphs? What does the value of "a" do to the graph?

Name the amplitude of each graph.

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As shown in the graph below, the graph of y = -3cos x is a reflection over the x-axis of the graph of y = 3cos x.

What is the amplitude of each function?y = 3cos x y = -3cos x

The domain of each function is the set of real numbers and the range is {x|-3 ≤ x ≤ 3}.

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Sketch each graph on the interval from 0 to 2 :

y = 4cos x y = -.25 sin x

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25 What is the amplitude of y = 3cos x ?

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26 What is the amplitude of y = 0.25cos x ?

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27 What is the amplitude of y = -s in x ?

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28 What is the range of the function y = 2s in x ?

A All real numbersB -2 < x < 2C 0 ≤ x ≤ 2D -2 ≤ x ≤ 2

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29 What is the domain of y = -3cos x ?

A All real numbersB -3 < x < 3C 0 ≤ x ≤ 3D -3 ≤ x ≤ 3

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30 Which graph represents the function y = -2s in x?

AB

C D

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31 What is the amplitude of the graph below?

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Predict, Explore, Confirm

1. Using your prior knowledge of transforming functions, predict what happens to the following functions:

2. Using your graphing calculator, insert the parent function into and the transformed function into . Compare the graphs.

3. Do your conclusions match your predictions?

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A periodic function is one that repeats its values at regular intervals. One complete repetition of the pattern is called a cycle. The period is the length of one complete cycle.

The trig functions are periodic functions.

The basic sine and cosine curves have a period of 2 , meaning that the graph completes one complete cycle in 2 units.

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y = s in bx or y = cos bx

Consider the graphs of y = cos x and y = cos 2x.

y = cos x

y = cos 2xone cycle

Notice that the graph of y = cos 2x completes one cycle twice as fast, or in units.

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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 a cycle in 4# . The period is 4# .

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The period for y = cos bx or y = s in bx is

2 b

y = cos x b = 1 P = = 2

y = cos 2x b = 2 P = =

y = cos 0.5x b = 0.5 P = = 4

P =

2 1

2

2

2

0.5

2

b2

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32 What is the period of

A

B

C

D

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33 What is the period of

A

B

C

D

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34 What is the period of

A

B

C

D

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Sketch the graph of each function from x = 0 to x = 2 .

y = 2cos 3x

y = -2cos 2xy = s in 2x

y = cos x

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35 What is the period of the graph below?

A

B 2

C 3

D

2

2

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36 What is the period of the graph shown?

A

B

C

D

2

3

23

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37 What is the equation of this function?

AB

C

D

y = sin 3xy = cos 3xy = 3cos xy = 3sin x

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Predict, Explore, Confirm

1. Using your prior knowledge of transforming functions, predict what happens to the following functions:

2. Using your graphing calculator, insert the parent function into and the transformed function into . Compare the graphs.

3. Do your conclusions match your predictions?

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Translating Sine and Cosine Functions

Trig functions can be translated in the same way as any other function.

The horizontal shift is called a phase shift.

What are your conclusions from the graphing calculator activity?

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k

Horizontal or phase shift

Vertical shift

Drag each equation to the matching graph

y = cos x

y = cos (x + )2

y = sin x + 2y = sin x

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Consider the graphs of

In order to determine the phase shift, the coefficient of x must be factored out.

In , the 2 is factored out. The phase shift is .

In , when the 2 is factored out, we get . The phase shift is .

and (which is which?)

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Another way to determine the phase shift is to set the quantity inside the parenthesis equal to 0 and solve for the variable.

Example: Set

Solve for x:

So, the phase shift is 2 .

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y= sin (x) + k or y= cos (x) + kVertical ShiftThe k moves the graph up or down.

The graph below is of the equation y = 2 sin (3x).

The midline of this graph is the horizontal line y = 0.

Sketch the graph of y = 2 sin (3x) + 1.

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42 What is the vertical shift in

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43 What is the vertical shift in

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44 What is the vertical shift in

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Graphing a Sine or Cosine Function:

Step 1: Identify the amplitude, period, phase shift and vertical shift.

Step 2: Draw the midline (y = k)

Step 3: Find 5 key points - maximums, minimums and points on the midline

Step 4: Draw the graph through the 5 points.

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Example:

Step 1: Amplitude: |-1| = 1

Period:

Phase Shift:

Vertical Shift: 2 (up 2)

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Step 2: Draw the midline y = 2Step 3: Find the 5 key points

Note: for x, adding comes from dividing the cycle, 3 by 4.

For y, adding and subtracting 1 comes from the amplitude.

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Step 4: Graph

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You try:

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49 What is the amplitude of this cosine graph?

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

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51 What is the vertical shift of this cosine graph?

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52 Which of the following of the following is an equation for the graph?

A

B

C

D

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The equation y = 4.2cos (π/6(x - 1)) + 13.7 can be used to model the average temperature of Wellington, NZ in degrees Celsius, where x represents the month, 1 - 12. Sketch the graph of this equation. What is the average temperature in June?

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Graphing the Tangent FunctionGraph by creating a table of values of key points. One option is to use the set of values for x that are multiples of , and the corresponding values of y or tan x.

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Notice that the tangent function is not defined for values of x where cos x = 0, or starting at and every units in either direction.

This is shown on the graph by the vertical lines, or asymptotes at these x values.

The period of the function is units, because there is one complete cycle from to . As x approaches or any odd multiple of from the left, y increases and approaches infinity. As x approaches from the right, y decreases and approaches negative infinity.

This is shown on the graph by the vertical lines, or asymptotes at these x values.

The period of the function is units, because there is one complete cycle from to . As x approaches or any odd multiple of from the left, y increases and approaches infinity. As x approaches from the right, y decreases and approaches negative infinity.

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Example: Sketch the graph of y = tan (x + ) + 2

Asymptotes will be at 0, , 2 , etc. The midline will be at y = 2.

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53 Which graph represents y = -tan x?

A B

C D

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Trigonometric Identities

Return toTable of

Contents

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Key Ideas

An identity is a mathematical equation that is true for all defined values of the variable.

A trigonometric identity is an identity that contains one or more trig ratios.

By contrast, a conditional equation is one that is only true for a limited set of values.

By learning trig identities, we will be able to replace complicated expressions with simpler ones to solve and verify more difficult equations and identities.

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Drag each equation into the correct box:

Identities Conditional Equations

3x + 4 = 93x + 4 = 3x + 4

sin # + cos # = 1

2x5 =x3

tan θ cot θ =1 2(x-1) = 2x - 2

(x + 3)2 = x2 + 9 sin 4x = 4sin x

5x - 7y = -(7y - 5x)

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Basic Trig Identities

csc # = 1 sin #

sec # = 1 cos #

cot # = 1 tan #

sin # = 1 csc # 1 1 cos # = 1

sec #tan # = 1

cot #

Reciprocal Identities

Tangent Identity

Cotangent Identity

tan # = sin # cos #

cot # = cos # sin #

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By using the basic identities we can change an expression into an equivalent expression.

Also, all of the rules of addition, subtraction, multiplication and division that we learned to solve equations and manipulate expressions can be applied to trig expressions and equations.

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Algebraic example Trig example

(x - y)(x + y) = x 2 - y2 (1 - cos #)(1 + cos #) = 1 - cos2#

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Recall the unit circle, x2 + y2 = 1.

1

(1,0)

(0,-1)

(0,1)

(-1,0)

(cos #,sin #)

Pythagorean Identities

For any point (x, y) on the circle, its coordinates are

(cos #, sin #).

Therefore,

(cos #)2 + (sin #)2 = 12, which is usually written as

cos2 θ + sin2 θ = 1

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Pythagorean Identities

How do we transform the first identity, which is derived from the unit circle, to the other two?

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Alternative Forms of Identities

Since we know that 3 + 5 = 8, we also know that

8 - 5 = 3 and 8 - 3 = 5. In elementary school we call these equivalent equations "fact families".

Similarly, if

cos2 θ + sin2 θ = 1, it follows that

1 - cos2 θ = sin2 θ and

1 - sin2 θ = cos2 θ.

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More Alternative Forms

Another fact family tells that since = 4, it follows that 4 ⋅ 5 = 20.

Since sec θ = , then sec θ cos θ = 1 (multiply both sides of the first equation by cos #).

205

1 cos θ

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Simplifying Trig Expressions

Example 1: Simplify csc θ cos θ tan θ.

Rewrite each trig ratio in terms of cos and sin:

1 sin #sin θ

⋅ cos θ ⋅ = 1cos #

(When multiplying fractions, it is often easier to reduce or cancel before you multiply.)

Example 2: Simplify csc2 θ(1 - cos2 θ). 1 sin2 θ ⋅ (sin2 θ) = 1

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Verifying an Identity

Transform one side of the identity to be the same as the other side

sin # cot # = cos # sin # ⋅ = cos # cos #

sin #

Example 1: Verify

Example 2: Verify

cos θ csc θ tan θ = 1

cos # ⋅ ⋅ = 1 1 sin #

sin # cos #

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Simplify:

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Simplify:

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Simplify:

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Simplify:

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Verify:

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Verify:

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54 Which equation is NOT an identity?

A sin2 x= 1 - cos2 xB 2 cot x = C tan2 x = sec2 x - 1D sin2 x = cos2 x - 1

2cos xsin x

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55 The following expression can be simplified to which choice?

A

B

C

D

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56 The following expression can be simplified to which choice?

A

B

C

D

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57 The following expression can be simplified to which choice?

A

B

C

D

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