Unit 7: Trigonometry – Part 1 · Ex. 2: Convert the following degree measure to radians: a) 400°...

100 AMHS Precalculus - Unit 7

Unit 7: Trigonometry – Part 1

Right Triangle Trigonometry

Hypotenuse Opposite α

Adjacent

a) Sine sin( )α =

b) Cosine cos( )α =

c) Tangent tan( )α =

d) Cosecant csc( )α =

e) Secant sec( )α =

f) Cotangent cot( )α =

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

θ 3

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

7

45˚


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˚


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?


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 α (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.


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) 56π

b) 10π

−

c) 4π

d) 3

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

a) 400° b) -120°

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

Ex. 3: Find a coterminal angle, one positive and one negative, to 53π

.


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(α ) =

2. cos(α )=

Ex. 1: Find the exact value

a) sin( )4π

b) cos( )2π

c) sin( )6π

d) cos( )3π

e) sin(150 )°

f) 11cos( )6π

g) 3sin( )2π

h) 5cos( )3π

i) sin(330 )°

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

a) 13sin( )4π

b) 7sin( )6π

− c) sin( 300 )− °


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

cos( ) cos( )α α− = sin( ) sin( )α α− = −

Ex. 3: Find the exact value.

a) 7cos( )6π

− b) 3sin( )4π

− c) 13cos( )4π

−

Reference Angles

For any angle α in standard position, the reference angle ( 'α ) associated with α is the acute angle formed by the terminal side of α and the x - axis.

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

a) 23πα = b) 2.3α = c) 5

4πα = d) 5

3πα =


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) 2sin( )3π

b) 5cos( )6π

c) 5sin( )3π

d) cos( )3π

− e) cos( 300 )− ° f) sin(150 )°

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

a) 2sin( )2

θ = b) 3cos( )2

θ = −

Ex. 5: If 2sin( )3

t = − and 32

t ππ ≤ ≤ , find the value of cos( )t .


Arc Length

In a circle of radius r , the length s of an arc with angle θ radians is: s rθ=

Ex. 1: Find the length of an arc of a circle with radius 5 and an angle54π

.

Ex. 2: Find the length of an arc of a circle with radius 13 and an angle 30° .

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

Area of a Circular Sector

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

212

A r θ=

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


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?


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 π= −

d) ( ) sin(2 )f x x=


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 π= −

Amplitude: Period: Horizontal Shift: End Points:

b) 1( ) 3sin( )2 4

f x x π= +



c) ( ) 1 2cos(2 )4

f x x π= + −


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

a) x -scale is 4π

b) x -scale is π

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

Unit 7: Trigonometry – Part 1 · Ex. 2: Convert the following degree measure to radians: a) 400°...

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Transcript of Unit 7: Trigonometry – Part 1 · Ex. 2: Convert the following degree measure to radians: a) 400°...