Name ____________________________________________ Period



Name ___________________________________________________________ Period ___________

Precalculus: Trigonometric Functions

4.1: Angles and Their Measures

Example 1: Convert the radians to degrees and the degrees to radians.

a) 75° b) 30° c) [pic] d) [pic]

Example 2: Working with DMS Measure

a) Convert 37.425° to DMS b) Convert 42°24’36’’ to degrees

Circular Arc Length

Example 3: Finding Arc Length

a) Find the perimeter of a 60° slice of a large pizza.

b) The running lanes at the Emery Sears track at Bluffton College are 1 meter wide. The inside radius of lane 1 is 33 meters and the inside radius of lane 2 is 34 meters. How much longer is lane 2 than lane 1 around one turn?

[pic]

Example 4: Angular and Linear Motion

a) Albert Juarez’s truck has wheels 36 inches in diameter. If the wheels are rotating at 630 rpm (revolutions per minute), find the truck’s speed in miles per hour.

4.2: Trigonometric Functions of Acute Angles

Right Triangle Trigonometry

Example 1: Find the six trig functions of the angle [pic].

a)

Example 2: Assume that [pic] is an acute angle in a right triangle satisfying the given conditions. Evaluate the remaining trigonometric functions.

a) [pic]

Complete the table of values below. You will want to memorize these trig values.

Example 2: Answer the following without using a calculator.

a) [pic] b) [pic] c) [pic]

Example 3: Find [pic] with using a calculator.

a) [pic] b) [pic] c) [pic]

Applications of Right Triangle Trigonometry

Example 4:

A right triangle with a hypotenuse of 8 includes a 37° angle (see below). Find the measure of the other angle and the lengths of the other two sides.

Example 5:

From a point 340 feet away from the base of the Peachtree Center Plaza in Atlanta Georgia, the angle of elevation to the top of the building is 65°. Find the height h of the building.

Example 5:

Find x.

4.3: Trigonometry Extended: The Circular Functions

Coterminal Angles: Angles that have the same initial and terminal side.

All three angles below are coterminal.

[pic] [pic] [pic]

Positive coterminal angle Negative coterminal angle Positive coterminal angle

Example 1: Finding Coterminal Angles

Find and draw a positive angle and a negative angle that are coterminal with the given angle.

a) 30° b) −150° c) [pic]

Example 2: Evaluating Trig Functions Determined by a Point

Let [pic] be the acute angle in standard position whose terminal side contains the given point. Find the six trigonometric functions of [pic].

a) (5, 3) b) (8, 11) c) (5, −3)

[pic] [pic] [pic]

Example 3: Evaluating Trig Functions

Find the following without using a calculator.

a) [pic] b) [pic]

[pic] [pic]

c) [pic] d) [pic]

[pic] [pic]

Example 4: Evaluating Trig Functions of Quadrantal Angles

Find each of the following, if it exists. If the value is undefined write “undefined”.

a) [pic]

b) [pic]

c) [pic]

Example 5: Using one trig ratio to find the others

Find cos[pic] and tan[pic] by using the given information to construct a reference triangle.

a) [pic] and [pic] b) [pic] and [pic]

4.4: Graphs of Sine and Cosine: Sinusoids

Transformations of Sine and Cosine Graphs

Example 1: Changing Amplitude

Find the amplitude of each function and use it to sketch the graph.

a) [pic] b) [pic]

[pic] [pic]

Example 2: Horizontal Stretch or Shrink and Period Change

Find the period of each function and use the language of transformations to describe how the graphs are related.

a) [pic] b) [pic]

[pic] [pic]

Example 3: Finding the Frequency

Find the frequency of the function and interpret its meaning graphically. Graph the function.

a) [pic]

Example 4: Graphing Transformations

Graph each sine or cosine function.

a) [pic] b) [pic]

c) [pic] d) [pic]

Example 5: Writing equations of Sine and Cosine graphs

a) amplitude = 2, period = [pic], point (0, 0) b) amplitude = 3, period = [pic], point (0, -3)

4.5: Graphs of Tangent, Cotangent, Secant, and Cosecant

Since [pic], [pic]has vertical asymptotes everywhere [pic]

Example 1: Graphing Tangent Functions

Locate the vertical asymptotes and graph two periods of the function. Describe the transformations.

a) [pic] b) [pic]

[pic] [pic]

Example 1: Graphing Cotangent Functions

Locate the vertical asymptotes and graph two periods of the function. Describe the transformations.

a) [pic] b) [pic]

[pic] [pic]

Example 3: Graphing Secant Functions

Locate the vertical asymptotes and graph two periods of the function. Describe the transformations.

a) [pic] b) [pic]

[pic] [pic]

Example 3: Graphing Cosecant Functions

Locate the vertical asymptotes and graph two periods of the function. Describe the transformations.

a) [pic] b) [pic]

[pic] [pic]

Solving a Trigonometric Equation Algebraically

Example 4: Solve for x in the given interval without using a calculator.

a) [pic] b) [pic]

4.7: Inverse Trigonometric Functions

Remember inverse functions are reflections of the graph over the line y = x.

1. Sketch a graph of [pic] on top of the sin x graph.

2. Is the inverse graph a function?

3. The restrictions on the domain for inverse functions exist in order to make them be functions (pass the vertical line test).

Example 1: Evaluating Inverse Functions Without a Calculator

a) [pic] b) [pic]

c) [pic] d) [pic]

Example 2: Evaluating Inverse Functions Without a Calculator

a) [pic] b) [pic]

c) [pic] d) [pic]

Example 3: Evaluating Inverse Functions Using a Calculator

Use a calculator to find the approximate value. Express your answer degrees on example (a) and in radians on example (b).

a) [pic] b) [pic]

4.8: Solving Problems with Trigonometry

Example 1: Using Angle of Depression

The angle of depression of a buoy from the top of the

Barnegat Bay lighthouse 130 feet above the surface of

the water is [pic]. Find the distance x from the base of the

lighthouse to the buoy.

Example 2: Making Indirect Measurements

From the top of the 100-ft-tall Altgelt Hall a man

observes a car moving toward the building. If the

angle of depression of the car changes from 22° to

46° during the period of observation, how far does

the car travel?

Example 3: Finding Height Above Ground

A large, helium-filled penguin is moored at the beginning of a

parade route awaiting the start of the parade. Two cables

attached to the underside of the penguin make angles of [pic]

and [pic] with the ground and are in the same plane as a

perpendicular line from the penguin to the ground. If the

cables are attached to the ground 10 feet from each other,

how high above the ground is the penguin?

Example 4: Using Trig in Navigation

A U.S. Coast Guard patrol boat leaves Port Cleveland and averages 35 knots (nautical mph) traveling for 2 hours on a course of 53° and then 3 hours on a course of 143°. What is the boat’s bearing and distance from Port Cleveland?

Example 5: Calculating Harmonic Motion

In a mechanical linkage a wheel with an 8 cm radius turns with an angular velocity of [pic] radians/sec.

a) What is the frequency of the piston?

b) What is the distance from the starting position (t = 0) exactly 3.45 seconds after starting?

Example 6: Calculating Harmonic Motion

A mass oscillating up and down on the bottom of a spring (assuming perfect elasticity and no friction or air resistance) can be modeled as harmonic motion. If the weight is displaced a maximum of 5 cm, find the modeling equation if it takes 2 seconds to complete one cycle.

-----------------------

Degrees Radians

[pic]

[pic]

[pic]

[pic]

Conversion:

Radians to degrees:

[pic]

Degrees to radians:

[pic]

DMS (degree-minute-second)

In the DMS system of angular measure, each degree is subdivided into 60 minutes (denoted by ‘) and each minute is subdivided into 60 seconds (denoted by ‘’).

Arc Length Formula (Radian Measure)

[pic]

[pic]

[pic]

[pic] is in degrees

[pic]

Trigonometric Functions

[pic] [pic]

[pic] [pic]

[pic] [pic]

[pic]

adjacent

opposite

hypotenuse

[pic]

3

4

5

Special Right Triangles

30-60-90 45-45-90

30°

60°

1

[pic]

[pic]

45°

45°

1

[pic]

[pic]

| |30° = [pic] |45° = [pic] |60° = [pic] |

|sin[pic] | | | |

|cos[pic] | | | |

|tan[pic] | | | |

b

a

8

37°

65°

340 ft

h

x

7

42°

460°

-260°

100°

Trigonometric Functions of any Angle

[pic]

[pic] [pic]

[pic] [pic]

[pic] [pic]

[pic]

Unit Circle

The Sine Function

[pic]

[pic]

Domain:

Range:

Continuity:

Symmetry:

Boundedness:

Extrema:

Asymptotes:

End Behavior:

[pic] =

[pic] =

Periodic with period [pic]

The Cosine Function

Domain:

Range:

Continuity:

Symmetry:

Boundedness:

Extrema:

Asymptotes:

End Behavior:

[pic] =

[pic] =

Periodic with period [pic]

[pic]

[pic]

[pic] or [pic]

[pic] = amplitude b effects the period and frequency of the graph,

new period = [pic], frequency = [pic]

c = horizontal shift d = vertical shift

[pic]

[pic]

[pic]

[pic]

[pic]

The Tangent Function

[pic]

Domain: All reals except odd multiples of [pic]

[pic]

Range: [pic]

Continuous on its domain

Increasing on each interval of the domain

Symmetric with respect to the origin (odd)

Not bounded above or below

No local extrema

No horizontal asymptotes

Vertical asymptotes: x = [pic] for all odd integers k

End Behavior: [pic] and [pic]

Do Not Exist (DNE)

Periodic with period [pic]

[pic]

The Cotangent Function

[pic]

[pic]

[pic]

Vertical asymptotes everywhere [pic]

[pic]

[pic]

Vertical asymptotes everywhere [pic]

[pic]

[pic]

Vertical asymptotes everywhere [pic]

[pic]

[pic]

Inverse Cosine Function (Arccosine Function)

[pic]

Domain: [pic]

Range: [pic]

[pic]

Inverse Sine Function (Arcsine Function)

[pic]

Domain: [pic]

Range: [pic]

[pic]

Inverse Tangent Function (Arctangent Function)

[pic]

Domain: [pic]

Range: [pic]

[pic]

The following equation are always true whenever they are defined:

[pic] [pic] [pic]

On the other hand, the following equations are only true for x values in the “restricted” domain of sin, cos, and tan:

[pic]- 0. The motion has frequency [pic], which is the number of oscillations per unit of time.

[pic]

[pic]

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download