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) d)
Example 2: Working with DMS Measure
a) Convert 37.425° to DMSb) 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?
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 .
a)
Example 2: Assume that is an acute angle in a right triangle satisfying the given conditions. Evaluate the remaining trigonometric functions.
a)
Complete the table of values below. You will want to memorizethese trig values.
Example 2: Answer the following without using a calculator.
a) b) c)
Example3: Find with using a calculator.
a) b) c)
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.
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)
Example 2: Evaluating Trig Functions Determined by a Point
Let be the acute angle in standard position whose terminal side contains the given point. Find the six trigonometric functions of .
a) (5, 3) b) (8, 11) c) (5, −3)
Example 3: Evaluating Trig Functions
Find the following without using a calculator.
a) b)
c) d)
Example 4: Evaluating Trig Functions of Quadrantal Angles
Find each of the following, if it exists. If the value is undefined write “undefined”.
a)
b)
c)
Example 5: Using one trig ratio to find the others
Find cos and tan by using the given information to construct a reference triangle.
a) and b) and
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) b)
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) b)
Example 3: Finding the Frequency
Find the frequency of the function and interpret its meaning graphically. Graph the function.
a)
Example 4: Graphing Transformations
Graph each sine or cosine function.
a) b)
c) d)
Example 5: Writing equations of Sine and Cosine graphs
a) amplitude = 2, period = , point (0, 0)b) amplitude = 3, period = , point (0, -3)
4.5: Graphs of Tangent, Cotangent, Secant, and Cosecant
Since , has vertical asymptotes everywhere
Example 1: Graphing Tangent Functions
Locate the vertical asymptotes and graph two periods of the function.Describe the transformations.
a) b)
Example 1: Graphing Cotangent Functions
Locate the vertical asymptotes and graph two periods of the function. Describe the transformations.
a) b)
Example 3: Graphing Secant Functions
Locate the vertical asymptotes and graph two periods of the function. Describe the transformations.
a) b)
Example 3: Graphing Cosecant Functions
Locate the vertical asymptotes and graph two periods of the function. Describe the transformations.
a) b)
Solving a Trigonometric Equation Algebraically
Example 4: Solve for x in the given interval without using a calculator.
a) b)
4.7: Inverse Trigonometric Functions
Remember inverse functions are reflections of the graph over the line y = x.
1. Sketch a graph of 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)b)
c)d)
Example 2: Evaluating Inverse Functions Without a Calculator
a)b)
c)d)
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)b)
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 . 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
and 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 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.
1