94 AMHS Precalculus - Unit 7

Unit 7: Trigonometry – Part 1

Right Triangle Trigonometry

Hypotenuse Opposite

Adjacent

a) Sine

sin( )

b) Cosine

cos( )

c) Tangent

tan( )

d) Cosecant

csc( )

e) Secant

sec( )

f) Cotangent

cot( )

Ex. 1: Find the values of the six trigonometric functions of the angle .

3

Ex.2: Find the exact values of the sin,cos, and tan of 45

7

45˚

95 AMHS Precalculus - Unit 7

Ex. 3: Find the exact values of the sin,cos, and tan of 60 and 30

Ex. 4: Find the exact value of x (without a calculator).

5

x

Ex. 5: Find all missing sides and angles (with a calculator).

21

30˚

33˚

30˚

96 AMHS Precalculus - Unit 7

Applications

An angle of elevation and an angle of depression can be measured from a point of reference and a

horizontal line. Draw two figures to illustrate.

Ex. 6: (use a calculator) A surveyor is standing 50 feet from the base of a large building. The surveyor

measures the angle of elevation to the top of the building to be 71.5˚. How tall is the building? Draw a

picture.

Ex. 7: A ladder leaning against a house forms a 67˚angle with the ground and needs to reach a window

17 feet above the ground. How long must the ladder be?

97 AMHS Precalculus - Unit 7

Angles – Degrees and Radians.

An angle consists of two rays that originate at a common point called the vertex. One of the rays is

called the initial side of the angle and the other ray is called the terminal side.

Angles that share the same initial side and terminal side are said to be coterminal.

To find a co-terminal angle to some angle (in degrees):

Radians – the other angle measure.

One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius

of the circle. Radians are a dimensionless angle measure.

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Conversion between degrees and radians

360 = radians.

Therefore,

a) 1 radian = degrees

b) 1 = radians

Ex. 1: Convert the following radian measure to degrees:

a) 5

6

b)

10

c) 4

d) 3

Ex. 2: Convert the following degree measure to radians:

a) 400 b) -120

To find a co-terminal angle to some angle (in radians):

Ex. 3: Find a coterminal angle, one positive and one negative, to 5

3

.

99 AMHS Precalculus - Unit 7

100 AMHS Precalculus - Unit 7

We use the unit circle to quickly evaluate the trigonometric functions of the common angle found on it.

To summarize how to evaluate the Sine and Cosine of the angles found on the unit circle:

1. sin( ) =

2. cos( )=

Ex. 1: Find the exact value

a) sin( )4

b) cos( )

2

c) sin( )

6

d) cos( )3

e) sin(150 )

f) 11

cos( )6

g) 3

sin( )2

h)

5cos( )

3

i) sin(330 )

Ex. 2: Find the exact value by finding coterminal angles that are on the unit circle.

a) 13

sin( )4

b)

7sin( )

6

c) sin( 300 )

101 AMHS Precalculus - Unit 7

Cosine is an even function. Sine is an odd function.

cos( ) cos( ) sin( ) sin( )

Ex. 3: Find the exact value.

a) 7

cos( )6

b)

3sin( )

4

c)

13cos( )

4

Reference Angles

For any angle in standard position, the reference angle ( ' ) associated with is the acute angle

formed by the terminal side of and the x - axis.

Ex. 1: Find the reference angle ' for the given angles.

a) 2

3

b) 2.3 c)

5

4

d)

5

3

102 AMHS Precalculus - Unit 7

The signs (+ or – value) of the Sine, Cosine and Tangent functions in the four quadrants of the Euclidean

plane can be summarized in this way:

Ex.2: Find the exact value.

a) 2

sin( )3

b) 5

cos( )6

c)

5sin( )

3

d) cos( )3

e) cos( 300 ) f) sin(150 )

Ex.3: Find all values of in the interval [0,2 ] that satisfy the given equation

a) 2

sin( )2

b) 3

cos( )2

Ex. 5: If 2

sin( )3

t and 3

2t

, find the value of cos( )t .

103 AMHS Precalculus - Unit 7

Arc Length

In a circle of radius r , the length s of an arc with

angle radians is: s r

Ex. 1: Find the length of an arc of a circle with radius 5 and an angle5

4

.

Ex. 2: Find the length of an arc of a circle with radius 13 and an angle30 .

Ex. 3: The arc of a circle of radius 3 associated with angle has length 5. What is the measure of ?

Area of a Circular Sector

In a circle of radius r , the area A of a circular sector formed by an angle of radians is

21

2A r

Ex. 1: Find the area A of a sector with angle 45 in a circle of radius 4.

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Graphs of the Sine and Cosine Functions

Ex. 1: Graph ( ) sinf x x

Domain: Range: x -intercepts: Period: Amplitude: Even or odd?

Ex. 2: Graph ( ) cosf x x

Domain: Range: x -intercepts: Period: Amplitude: Even or odd?

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Ex.3: Graph one period of each function

a) ( ) 2cos( )f x x b) ( ) 1 sin( )f x x

c) ( ) cos( )2

f x x

d) ( ) sin(2 )f x x

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Graphs of ( ) sin( )f x A Bx C D and ( ) cos( )f x A Bx C D where 0A and 0B have:

Amplitude:

Period:

Horizontal shift (Phase shift):

Vertical shift:

Ex.1: Graph one period of each function.

a) ( ) 2sin( )3

f x x

Amplitude: Period: Horizontal Shift: End Points:

b) 1

( ) 3sin( )2 4

f x x

Amplitude: Period: Horizontal Shift: End Points:

107 AMHS Precalculus - Unit 7

c) ( ) 1 2cos(2 )4

f x x

Amplitude: Period: Horizontal Shift: End Points:

Ex. 2: Write a Sine or Cosine function whose graph matches the given curve.

a) x -scale is 4

b) x -scale is

Ex. 3: Write a Sine and Cosine function whose graph matches the given curve. x -scale is 3