chapter 4 trigonometry -...

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Chapter 4 Trigonometry Section 4.1 Radian and Degree Measure Objective: In this lesson you learned how to describe an angle and to

convert between radian and degree measure. I A

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Important Vocabulary Define each term or concept. Trigonometry aaa aaaaa aaaa aaa aaaaaaaaaaaa aa aaaaaaaaaaa Central angle of a circle aa aaaaa aaaaa aaaaaa aa aaa aaaaaa aa aaa aaaaaaa Complementary angles aaa aaaaaaaa aaaaaa aaaaa aaa aa aaa aaaaaaa aa aaaa Supplementary angles aaa aaaaaaaa aaaaaa aaaaa aaa aa a aaaaaaa aa aaaaa Degree aaa aaaa aaaaaa aaaa aa aaaaa aaaaaaaa aaaaaaa aa aaa aaaaaa aa a aaaaaaa aa aaa aaaaaa aaaa aa aaaaaaaaaa aa a aaaaaaaa aa aaaaa aa a aaaaaaaa aaaaaaaaaa aaaaa aaa aaaaaa aa aa aaaaaa

. Angles (Page 282) What you should learn How to describe angles

n angle is determined by . . . aaaaaaaa a aaa aaaaaaaaaaa

aaaa aaa aaaaaaaaa

he initial side of an angle is . . . aaa aaaaaaaa aaaaaaaa aa

aa aaaaaaa aaa aa aaa aaaaaaaaa aa aa aaaaaa

he terminal side of an angle is . . . aaa aaaaaaaa aa aaa

aa aaaaa aaa aaaaaaaa aaaa aa aaaaa aa aaaaaaa

he vertex of an angle is . . . aaa aaaaaaaa aa aaa aaa aaaa

a aaa aaaaaaaaa aa aa aaaaaa

n angle is in standard position when . . . aaa aaaaaaa

aaaaa aa aa aaa aaaaaa aa a aaaaaaaaaa aaaaaa aaa aaa aaaaaaa

aaa aaaaaaaaa aaaa aaa aaaaaaaa aaaaaaa

positive angle is generated by a aaaaaaaaaaaaaaa a

otation; whereas a negative angle is generated by a

aaaaaaaaa a rotation.

f two angles are coterminal, then they have . . . aaa aaaa

aaaaaa aaaa aaa aaa aaaa aaaaaaaa aaaaa

arson/Hostetler Precalculus/Precalculus with Limits Notetaking Guide opyright © Houghton Mifflin Company. All rights reserved. 73

74 Chapter 4 • Trigonometry

What you should learn How to use radian measure

II. Radian Measure (Pages 283−285) The measure of an angle is determined by . . . aaa aaaaaa aa

aaaaaaaa aaaa aaa aaaaaaa aaaa aa aaa aaaaaaaa aaaaa

One radian is the measure of a central angle θ that . . .

aaaaaaaaaa aa aaa a aaaaa aa aaaaaa aa aaa aaaaaa a aa aaa

aaaaaaa

A central angle of one full revolution (counterclockwise)

corresponds to an arc length of s = aaa .

A full revolution around a circle of radius r corresponds to an

angle of aa radians. A half revolution around a circle

of radius r corresponds to an angle of a a radians.

Angles with measures between 0 and π/2 radians are

aaaaa angles. Angles with measures between π/2

and π radians are aaaaaa angles.

To find an angle that is coterminal to a given angle θ, . . .

aaa aa aaaaaaaa aa aaa aaaaaaa aaaaaaaaa aa aaa aa aaa aaaaaaa

aa aa

Example 1: Find an angle that is coterminal with θ = − π/8. aaaaa Example 2: Find the supplement of θ = π/4. aaaa III. Degree Measure (Pages 285−286)

What you should learn How to use degree measure

A full revolution (counterclockwise) around a circle corresponds

to aaa degrees. A half revolution around a circle

corresponds to aaa degrees.

Larson/Hostetler Precalculus/Precalculus with Limits Notetaking Guide Copyright © Houghton Mifflin Company. All rights reserved.

Section 4.1 • Radian and Degree Measure 75

To convert degrees to radians, . . . aaaaaaaa aaaaaaa aa

aa aaaaaaaaaa

To convert radians to degrees, . . . aaaaaaaa aaaaaaa aa

aaaaaaa aaaaa

Example 3: Convert 120° to radians. aaaa Example 4: Convert 9π/8 radians to degrees. aaaaaa Example 5: Complete the following table of equivalent degree

and radian measures for common angles.

IV. Applications of Angles (Pages 287−289) For a circle of radius r, a central angle θ intercepts an arc length

s given by a a aa where θ is measured in radians.

Note that if r = 1, then s = θ , and the radian measure of θ equals

aaa aaa aaaaaa .

Consider a particle moving at constant speed along a circular arc of radius r. If s is the length of the arc traveled in time t, then the linear speed of the particle is

linear speed = aaaa aaaaaaaaaaaaaa a aaa a If θ is the angle (in radian measure) corresponding to the arc length s, then the angular speed of the particle is

angular speed = aaaaaaaa aaaaaaaaaaaaa a aaa a Example 6: A 6-inch-diameter gear makes 2.5 revolutions per

second. Find the angular speed of the gear in radians per second.

aa aaaaaaa aaa aaaaaa


What you should learn How to use angles to model and solve real-lifeproblems

θ (degrees) 0° aaa 45° aaa 90° aaaa 270° θ (radians) a π/6 aaa π/3 aaa π aaaa


A sector of a circle is . . . aaa aaaaaa aaaaaaa aa aaa aaaaa aa

aaa aaaaaa aaa aaaaa aaaaaaaaaaa aaaa

For a circle of radius r, the area A of a sector of the circle with

central angle θ is given by ___________ , where

θ is measured in radians.

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Homework Assignment Page(s) Exercises


Section 4.2 • Trigonometric Functions: The Unit Circle 77 Name______________________________________________

Section 4.2 Trigonometric Functions: The Unit Circle Objective: In this lesson you learned how to identify a unit circle and its

relationship to real numbers.

Important Vocabulary Define each term or concept. Unit circle a aaaaaa aa aaaaaa a aaaaaaaa aa aaa aaaaaa aaa aaaaa aa aaa aaaaaaaa aa a a a aa aaa aaaaa aaa aaa aa aaa aaaaaaaaaaaa aa aaaaaaaaaaaaa

a

a

Period a aaaaaaaa a aa aaaaaaaa aa aaaaa aaaaaa a aaaaaaaa aaaa aaaaaa a aaaa aaaa a aaa a aa a aaaa aaa aaa a aa aaa aaaaaa aa aa aaa aaaaaaaa aaaaaa a aaa aaaaa a aa aaaaaaaa aa aaaaaa aaa aaaaaa aa aa

I. The Unit Circle (Page 294) What you should learn How to identify a unit circle and its relationship to real numbers

As the real number line is wrapped around the unit circle, each

real number t corresponds to . . . a aaaaa aaa aa aa aaa aaaa

The real number 2π corresponds to the point aa aa a

on the unit circle.

Each real number t also corresponds to a aaaaaaa aaa a a

(in standard position) whose radian measure is t. With this

interpretation of t, the arc length formula s = rθ (with r = 1)

indicates that . . . aaa aaaa aaaaaa a aa aaa aaaaaa aa aaa

aaa aaaaaaaaaaa aa aaa aaaaa aa aaaaa aa aaaaaaaa

What you should learn How to evaluate trigonometric functions using the unit circle

II. The Trigonometric Functions (Pages 295−297) The coordinates x and y are two functions of the real variable t. These coordinates can be used to define six trigonometric functions of t. List the abbreviation for each trigonometric function. Sine aaa a Cosecant aaa a Cosine aaa a Secant aaa a Tangent aaa a Cotangent aaa a



Let t be a real number and let (x, y) be the point on the unit circle corresponding to t. Complete the following definitions of the trigonometric functions: sin t = a a cos t = a a tan t = aaaa a a a a cot t = aaaa a a a a sec t = aaaa a a a a csc t = aaaa a a a a The cosecant function is the reciprocal of the aaaa a

function. The cotangent function is the reciprocal of the

aaaaaaa function. The secant function is the

reciprocal of the aaaaaa function.

Complete the following table showing the correspondence between the real number t and the point (x, y) on the unit circle when the unit circle is divided into eight equal arcs.

t 0 π/4 π/2 3π/4 π 5π/4 3π/2 7π/4 x a aa aa a a aa aa a a a aa aa a aa aa

y a aa aa a aa aa a a aaaa a a a aa aa
Complete the following table showing the correspondence between the real number t and the point (x, y) on the unit circle when the unit circle is divided into 12 equal arcs. E
T

a

t 0 π/6 π/3 π/2 2π/3 5π/6 π 7π/6 4π/3 3π/2 5π/3 11π/6x a aa aa aaa a a aaa a aa aa a a a aa aa a aaa a aaa aa aa

y a aaa aa aa a aa aa aaa a a aaa a aa aa a a a aa aa a aaa

xample 1: Find the following:

(a) 3

cosπ (b) 4

3tan π (c) 6

7csc π

aaa aaa aaaaa a aaaaaa a a

What you should learn How to use the domain and period to evaluate sine and cosine functions

III. Domain and Period of Sine and Cosine (Pages 297−298)

he sine function’s domain is aaa aaa aa aaa aa aaaaaaa ,

nd its range is aa aa aa .


Section 4.2 • Trigonometric Functions: The Unit Circle 79 Name______________________________________________

The cosine function’s domain is aaa aaa aa aaa aaaa aaaaa ,

and its range is aa aa aa .

The period of the sine function is aa . The

period of the cosine function is aa .

Which trigonometric functions are even functions?

aaaaaa aaa aaaaaa a

Which trigonometric functions are odd functions?

aaaaa aaaaaaaaa aaaaaaaa aaa aaaaaaaaa a

Example 2: Evaluate 6

31sin π

a aaa IV. Evaluating Trigonometric Functions with a Calculator What you should learn

How to use a calculator to evaluate trigonometric functions

(Page 298) To evaluate the secant function with a calculator, . . .

aaaaaaaa aaa aaaaaaaaaa aaaaa aaaaaaa aaa aaaa aaa aaa a aaaaaa

Example 3: Use a calculator to evaluate (a) tan 4π/3, and

(b) cos 3. aaa aaaaaaaaaaa aaa a aaaaaaaaaaaa



Additional notes

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Section 4.3 • Right Triangle Trigonometry 81 Name______________________________________________

Section 4.3 Right Triangle Trigonometry Objective: In this lesson you learned how to evaluate trigonometric

functions of acute angles and how to use the fundamental trigonometric identities.

What you should learn How to evaluate trigonometric functions of acute angles

I. The Six Trigonometric Functions (Pages 301−303) In the right triangle shown below, label the three sides of the triangle relative to the angle labeled θ as (a) the hypotenuse, (b) the opposite side, and (c) the adjacent side.

θ

Let θ be an acute angle of a right triangle. Define the six trigonometric functions of the angle θ using opp = the length of the side opposite θ, adj = the length of the side adjacent to θ, and hyp = the length of the hypotenuse. sin θ = aaaaaaa a cos θ = aaaaaaa a tan θ = aaaaaaa a csc θ = aaaaaaa a sec θ = aaaaaaa a cot θ = aaaaaaa a The cosecant function is the reciprocal of the aaaa a

function. The cotangent function is the reciprocal of the

aaaaaaa function. The secant function is the

reciprocal of the aaaaaa function.

Example 1: In the right triangle below, find sin θ, cos θ, and

tan θ.

θ

Hypotenuse

12 aaa a a aaaaaa aaa a a aaaaa aaa a a aaaaa

5



Give the sines, cosines, and tangents of the following special angles:

==°6

sin30sin π aaa

==°6

cos30cos π aa aa

==°6

tan30tan π aa aa

==°4

sin45sin π aa aa

==°4

cos45cos π aa aa

==°4

tan45tan π a

==°3

sin60sin π aa aa

==°3

cos60cos π aaa

==°3

tan60tan π aa

Cofunctions of complementary angles are aaaaa . If θ

is an acute angle, then:

=−° )90sin( θ aaa a a =−° )90cos( θ aaa a a

=−° )90tan( θ aaa a a =−° )90cot( θ aaa a a

=−° )90sec( θ aaa a a =−° )90csc( θ aaa a a II. Trigonometric Identities (Pages 304−305)

What you should learn How to use the fundamental trigonometric identities

List six reciprocal identities: 1) aaa a a aaaaaa aa

2) aaa a a aaaaaa aa






Section 4.3 • Right Triangle Trigonometry 83 Name______________________________________________

List two quotient identities: 1) aaa a a aaaa aaaaaaa aa 2) aaa a a aaaa aaaaaaa aa List three Pythagorean identities: 1) aaa a a aaa a a aa a

2) a a aaa a a aaa aa a

3) a a aaa a a aaa aa a

III. Evaluating Trigonometric Functions with a Calculator What you should learn

How to use a calculator to evaluate trigonometric functions

(Page 305) To use a calculator to evaluate trigonometric functions of angles

measured in degrees, . . . aa aaaa aa aaaaa aaa aaa aaaaaaaaaa

aa aaaaaa aaaaa

Example 2: Use a calculator to evaluate (a) tan 35.4°, and

(b) cos 3.25° aaa aaaaaaaaaaaa aaa aaaaaaaaaaaa

What you should learn How to use trigonometric functions to model and solve real-life problems

IV. Applications Involving Right Triangles (Pages 306−307) What does it mean to “solve a right triangle?” aaaaaaa aaa aaa aaaaa aaa aaaa aa a aaaaa aaaaaaaa aaa aaa aa aaa aaaaa aaaaaa aaa aaa aaaaa aa aaaa aaa aa aaa aaaaa aaaaaa aa aaa aaa aaaaa aaa aaaaa aaa aaa aaaaa aa aaaa aaa aa aaa aaaaa aaaaaaa The term angle of elevation means . . . aaa aaaaa aaaa aaa

aaaaaaaaaa aaaaaa aa aa aaaaaaa

The term angle of depression means . . . aaa aaaaaaa aaaa aaa

aaaaa aaa aaaaaaaaaaa aaa aaaaa aaaa aaa aaaaaaaaaa aaaaaaaa

aa aa aaaaaaa



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Section 4.4 • Trigonometric Functions of Any Angle 85 Name______________________________________________

Section 4.4 Trigonometric Functions of Any Angle Objective: In this lesson you learned how to evaluate trigonometric

functions of any angle.

Important Vocabulary Define each term or concept. Reference angles aaa a aa aa aaaaa aa aaaaaaaa aaaaaaaaa aaa aaaaaaaaa aaaaa aa aaa aaaaa aaaaa a a aaaaaa aa aaa aaaaaaaa aaaa aa a aaa aaa aaaaaaaaaa aaaaa

What you should learn How to evaluate trigonometric functions of any angle

I. Introduction (Pages 312−313) Let θ be an angle in standard position with (x, y) a point on the

terminal side of θ and 022 ≠+= yxr . Complete the following definitions of the trigonometric functions of any angle: y

(x, y)

r

sin θ = aaa a cos θ = aaa a tan θ = aaaa a a a a cot θ = aaaa a a a a sec θ = aaaa a a a a csc θ = aaaa a a a a Name the quadrants in which the sine function is positive.

aaaaaaaaa a aaa aa a

Name the quadrants in which the sine function is negative.

aaaaaaaaa aaa aaa aa a

Name the quadrants in which the cosine function is positive.

aaaaaaaaa a aaa aa a

Name the quadrants in which the cosine function is negative.

aaaaaaaaa aa aaa aaa a

Name the quadrants in which the tangent function is positive.

aaaaaaaaa a aaa aaa a

Name the quadrants in which the tangent function is negative.

aaaaaaaaa aa aaa aa a

Example 1: If 2

1sin =θ and 0tan <θ , find θcos . a aaaaaa


θ
x


II. Reference Angles (Page 314) What you should learn How to use reference angles to evaluate trigonometric functions

Example 2: Find the reference angle θ ′ for (a) θ = 210° (b) θ = 4.1 aaa aaa aaa aaaaaa aaaaaaa III. Trigonometric Functions of Real Numbers What you should learn

How to evaluate trigonometric functions of real numbers

(Pages 315−317) To find the value of a trigonometric function of any angle θ, . . .

aaaaaaaaa aaa aaaaaaaa aaaaa aaa aaa aaaaaaaaaa aaaaaaaaa

aaaaa a aa aaaaaaaaa aa aaa aaaaaaaa aa aaaaa a aaaaa

aaaaa aaa aaaaaaaaaaa aaaa aa aaa aaaaaaaa aaaaaa

Example 3: Evaluate 6

11sin π .

a aaa Example 4: Evaluate . °240cos a aaa

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Section 4.5 • Graphs of Sine and Cosine Functions 87 Name______________________________________________

Section 4.5 Graphs of Sine and Cosine Functions Objective: In this lesson you learned how to sketch the graphs of sine

and cosine functions and translations of these functions. I F

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Important Vocabulary Define each term or concept. Amplitude aaaaaaaaaa aaaa aaa aaaaaaaa aaaaaaa aaa aaaaaaa aaa aaaaaaa aaaaaa aa a a a aaa a aa a a a aaa a aaa aa aaaaa aa aaaaaaaaa a a a aaPhase shift aaa aaaaaa aaa aa aaaaa aaa aaaaa aa a a a aaa aa aa aaaaaaa aa aaaaaa aaa aaaaa aa a a a aaaaaa a aaa

What you should learn How to sketch the graphs of basic sine and cosine functions

. Basic Sine and Cosine Curves (Pages 321−322)

or π20 ≤≤ x , the sine function has its maximum point at

aaaaa aa , its minimum point at aaaaa a aa ,

nd its intercepts at aaa aaa aaa aaaa aaaa aa .

or π20 ≤≤ x , the cosine function has its maximum points at

aaa aa aaa aaaa aa , its minimum point at aaa a a ,

nd its intercepts at aaaaa aa aaa aaaaaa aa .

xample 1: Sketch the basic sine curve on the interval [0, 2π]. y

x

xample 2: Sketch the basic cosine curve on the interval [0, 2π]. y

x

arson/Hostetler Precalculus/Precalculus with Limits Notetaking Guide opyright © Houghton Mifflin Company. All rights reserved.


II. Amplitude and Period (Pages 323−324) What you should learn How to use amplitude and period to help sketch the graphs of sine and cosine functions

The constant factor a in acts as . . . xay sin= a aaaaaaa

aaaaaaa a aaaaaaaa aaaaaaa aa a aaaaaaaa aaaaaa aa aaa aaaaa

aaaa aaaaaa

If 1>a , the basic sine curve is aaaaaaaaa . If

1<a , the basic sine curve is aaaaaa . The result is

that the graph of ranges between xay sin= a a aaa a a

instead of between − 1 and 1. The absolute value of a is the

aaaaaaaaa of the function . xay sin=

The graph of is a(n) xy sin5.0= aaaaaaaaaa in the

x-axis of the graph of . xy sin5.0−=

Let b be a positive real number. The period of bxay sin= and

is bxay cos= aaaa . If 0 < b < 1, the period of

is bxay sin= aaaaaaa than 2π and represents a

aaaaaaaaaa aaaaaaaaaa of the graph of xay sin= .

If b > 1, the period of is bxay sin= aaaa than

2π and represents a aaaaaaaaaa aaaaaaaaa of the

graph of . xay sin=

Example 3: Find the amplitude and the period of

. xy 3cos4−= aaaaaaaaa a aa aaaaaa a aaaa Example 4: Find the five key points (intercepts, maximum

points, and minimum points) of the graph of . xy 3cos4−=

aaa a aaa aaaaa aaa aaaaa aaa aaaa aaa aaaaaa a aa


Section 4.5 • Graphs of Sine and Cosine Functions 89 Name______________________________________________

What you should learn How to sketch translations of the graphs of sine and cosine functions

III. Translations of Sine and Cosine Curves (Pages 325−326) The constant c in the general equations )sin( cbxay −= and

creates . . . )cos( cbxay −= a aaaaaaaaaa aaaaaaaaaaa

aaaaaaa aa aaa aaaaa aaaa aaa aaaaaa aaaaaaa

Comparing with bxay sin= )sin( cbxay −= , the graph of

completes one cycle from )sin( cbxay −= aa a a a a to

aa a a a aa . By solving for x, the interval for one cycle

is found to be aaa to aaa a aaaa . This

implies that the graph of )sin( cbxay −= is the graph of

shifted by the amount bxay sin= aaa .

The period of the graph of )cos( cbxay −= is aaaa . Example 5: Find the amplitude, period, and phase shift of

)4/sin(2 π−= xy . aaaaaaaaa a aa aaaaaa a aaa aaaaa aaaaa a aaa Example 6: Find the five key points (intercepts, maximum

points, and minimum points) of the graph of )4/sin(2 π−= xy .

aaaaa aaa aaaaaa aaa a aaa aaaaaa a aaa aaaaaa aa IV. Mathematical Modeling (Page 327) What you should learn

How to use sine and cosine functions to model real-life data

Describe a real-life situation which can be modeled by a sine or cosine function. aaaaaaa aaaa aaaaa



Example 7: Find a trigonometric function to model the data in the following table.

Add

H Pa E

x 0 π/2 π 3π/2 2π y 2 4 2 0 2

a a a aaa a a a

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omework Assignment

ge(s)

xercises


Section 4.6 • Graphs of Other Trigonometric Functions 91 Name______________________________________________

Section 4.6 Graphs of Other Trigonometric Functions Objective: In this lesson you learned how to sketch the graphs of other

trigonometric functions. I B

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Important Vocabulary Define each term or concept. Damping factor aaaa a aaaaaaaaaaaaa aaaaaaaa aa aaaaaaaaaa aa aaaaaaa aaaaaaaaa aaaa aaaaa aaaaaaaa aa aaaaaa aaa aaaaaaa aaaaaaa

What you should learn How to sketch the graphs of tangent functions

. Graph of the Tangent Function (Pages 332−333)

ecause the tangent function is odd, the graph of xy tan= is

ymmetric with respect to the aaaaaa . The period of

he tangent function is a . On the interval [0, π], the

angent function is undefined, and thus has a vertical asymptote,

t x = aaa . The domain of the tangent function is

aaa a a aaa a aa , and the range of the

angent function is aa aa aa .

escribe how to sketch the graph of a function of the form . )tan( cbxay −=

aaaa aa aaaaaaaa aaa aaaaaa aaaa aaaaaaaa aaaaaaaaaa aaa aaaaaaaaaa aaa aaaaaaaaaaa aaaaaaaa aaaaaaaaaa aaa aa aaaaa a aaaaaaa aaa aaaaaaaaa aa a a a a aaa aaa aa a a a aaaa aaa aaaaaaa aaaaaaa aaa aaaaaaaaaaa aaaaaaaaaa aa aa aaaaaaaaaaa a aaa aaaaaa aaa aaaaaa aa aaa aaaaaaaa aaaaaaa aaa aaaaaaaaaa aaaaaaaa aaaaaaaaaaa aaaaa aaaaaaaa aaa aaaaaaaaa aaa aaa aaaaaaaaaaaa aaaa a aaa aaaaaaaaa aaaaaa aaaaaaa aaa aaa aaaaaaaaaa aaa aaaaaa aaa aaaaa aaaaaaaa aaaaaa aaa aa aaa aaaaaaaaaa aaaaaa aa aaa aaaa aa aaaaaa

I. Graph of the Cotangent Function (Page 334) What you should learn How to sketch the graphs of cotangent functions

he graph of xy cot= is symmetric with respect to the

aaaaaa . The period of the cotangent function is

a . On the interval (0, π], the cotangent function is

ndefined, and thus has a vertical asymptote, at x = a .

arson/Hostetler Precalculus/Precalculus with Limits Notetaking Guide opyright © Houghton Mifflin Company. All rights reserved.


The domain of the cotangent function is aa a a aa ,

and the range of the cotangent function is aa aa aa .

What you should learn How to sketch the graphs of secant and cosecant functions

III. Graphs of the Reciprocal Functions (Pages 335−336) At a given value of x, the y-coordinate of csc x is the reciprocal

of the y-coordinate of aaa a .

The graph of xy csc= is symmetric with respect to the

aaaaaa . The period of the cosecant function is

aa . On the interval (0, π], the cosecant function is

undefined, and thus has a vertical asymptote, at x = a .

The domain of the cosecant function is aaa a a aa ,

and the range of the cosecant function is aa aa a aa aaa aaa aa .

At a given value of x, the y-coordinate of sec x is the reciprocal

of the y-coordinate of aaa a .

The graph of xy sec= is symmetric with respect to the

aaaaaa . The period of the secant function is aa .

On the interval [0, π], the secant function is undefined, and thus

has a vertical asymptote, at x = aaa . The domain of

the secant function is aaa a a aaa a aa , and

the range of the secant function is aa aa a aa aaa aaa aa .

To sketch the graph of a secant or cosecant function, . . . aaa aaaaaa aaaaa aaaa a aaaaaa aa aaa aaaaaaaaaa aaaaaaaaa aaaa aaaa aaaaaaaaaaa aa aaa aaaaaaaaaaaaa aa aaaaaa aaaaaa aa aaa aaaaa aa aaa aaaaaa aa aaaaaaaa aaaaaaaaa aaa aaaaaaaaaaaa aa aaa aaaaaaaaaa aaaaaaaa aaaaaa aaa aaaaaaaa aaaaaaaaaa aa aaa aaaaaa aa aaaaaaaa aaaaaaaaa


Section 4.6 • Graphs of Other Trigonometric Functions 93 Name______________________________________________

What you should learn How to sketch the graphs of damped trigonometric functions

IV. Damped Trigonometric Graphs (Pages 337−338) Explain how to sketch the graph of the damped trigonometric function , where is the damping factor. xxfy cos)(= )(xf aaaaaaa aaaa aaaaa aaa aaaaaaaaa aaaaaa aaaa aaa a aaaa aa aaaaaaa aaaa aaaaaa aaa aaaaa aa a a aaa a aaaaaaaaa aaaaa aaaaaa aa aaaa aaa aaaaa aaaaaaa aaa aaaaaa aa aaa aaaaaaaa aaaa aaaaa aaaaaaaa aa aaaaaaaaaaaa aa a a aaa aa Additional notes



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Section 4.7 • Inverse Trigonometric Functions 95 Name______________________________________________

Section 4.7 Inverse Trigonometric Functions Objective: In this lesson you learned how to evaluate the inverse

trigonometric functions and compositions of trigonometric functions with inverse trigonometric functions.

What you should learn How to evaluate and graph the inverse sine function

I. Inverse Sine Function (Pages 343−344) The inverse sine function is defined by . . .

a a aaaaaa a aa aaa aaaa aa aaa a a aa aaaaa a a a a a a aaa

a aaa a a a aaaa

The domain of y = arcsin x is aa aa aa . The range of

y = arcsin x is aa aaaa aaaa .

Example 1: Find the exact value: )1(arcsin − . a aaa II. Other Inverse Trigonometric Functions (Pages 345−346) The inverse cosine function is defined by . . .


a a a a aa

The domain of y = arccos x is aa aa aa . The range of

y = arccos x is aaa aa .

Example 2: Find the exact value: 21arccos .

aaa The inverse tangent function is defined by . . .


a aaa a a a aaaa

The domain of y = arctan x is aa aa aa . The range of

y = arctan x is aa aaaa aaaa .


What you should learn How to evaluate and graph the other inverse trigonometric functions


Example 3: Find the exact value: )3arctan( . aaa Example 4: Use a calculator to approximate the value (if

possible). Round to four decimal places. (a) arcos 0.85 (b) arcsin 3.1415 aaa aaaaaa aaa aaa aaaaaaaa

What you should learn How to evaluate and graph the compositions of trigonometric functions

III. Compositions of Functions (Pages 347−348) State the Inverse Property for the Sine function. aa a a a a a a aaa a aaa a a a aaaa aaaaaaaaaaaaaaa aa a a aaa aaaaaaaaaa aa a aaa State the Inverse Property for the Cosine function. aa a a a a a a aaa a a a a aa aaaaaaaaaaaaaaa aa a a aaa aaaaaaaaaa aa a aaa State the Inverse Property for the Tangent function. aa a aa a aaaa aaaaaa aaa a aaa a a a aaaa aaaaaaaaaaaaaaa aa a a aaa aaaaaaaaaa aa a aaa The inverse properties do not apply for arbitrary values of x and

y. For example, the inverse property for the sine function is not

value for values of y outside the interval

aaaaaa aaaa .

Example 5: If possible, find the exact value: (a) arcsin(sin 3π/4) (b) cos(arccos 0) aaa aaa aaa a



Section 4.8 • Applications and Models 97 Name______________________________________________

Section 4.8 Applications and Models Objective: In this lesson you learned how to use trigonometric functions

to solve real-life problems.

Important Vocabulary Define each term or concept. Simple harmonic motion aaa aaaaaaaaaa aaaaaaaaaaaa aa aaaaaaaa aa aa aaaaaa aaaaa aaaaa aaaaaaaaaa aaaa aaaa aaa aaaaaaaa aaaaaaa aaa aaaaaaa aaaaaa aaa aa aaaaaaaaa aa a aaaa aa aaaaaa aaaaaaaaa

What you should learn How to solve real-life problems involving right triangles

I. Applications Involving Right Triangles (Pages 353−354) Example 1: A ladder leaning against a house reaches 24 feet

up the side of the house. The ladder makes a 60° angle with the ground. How far is the base of the ladder from the house? Round your answer to two decimal places.

aaaaa aaaa II. Trigonometry and Bearings (Page 355) In surveying and navigation, a directional bearing measures . . .

aaa aaaaa aaaaa aaaa a aaaa aa aaaa aa aaaaa aaaaa aaaa a aaaaa

aaaaaaaaaaa aaaaa

The bearing N 70° E means . . . aa aaaaaaa aaaa aa aaaaaa Example 2: Write the bearing for the path shown in the

diagram below. N

30° W

S

a aaa a E


What you should learn How to solve real-life problems involving directional bearings


What you should learn How to solve real-life problems involving harmonic motion

III. Harmonic Motion (Pages 356−358) A point that moves on a coordinate line is said to be in simple

harmonic motion if . . . aaa aaaaaaaa a aaaa aaa aaaaaa aa

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aaa a aaa aaaa aaaaaaa aaaa aaaa a a aa

The simple harmonic motion has amplitude a a , period

aaaa , and frequency aaaaaa .

Example 3: Given the equation for simple harmonic motion

2sin3 td = , find:

(a) the maximum displacement, (b) the frequency of the simple harmonic motion,

and (c) the period of the simple harmonic motion.

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y

x

y

x

y

x



chapter 4 trigonometry -...

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