trigonometric functions chapter 5 texpoint fonts used in emf. read the texpoint manual before you...

Trigonometric Functions

Chapter 5

Angles and Their Measure

Section 5.1

Basic Terminology

Ray: A half-line starting at a vertex V

Angle: Two rays with a common vertex

Basic Terminology

Initial side and terminal side: The rays in an angleAngle shows direction and

amount of rotationLower-case Greek letters

denote angles

Basic Terminology

Positive angle: Counterclockwise rotation

Negative angle: Clockwise rotation

Coterminal angles: Share initial and terminal sides

Positive angle

Positive angle

Negative angle

Basic Terminology

Standard position: Vertex at originInitial side is positive x-axis

Basic Terminology

Quadrantal angle: Angle in standard position that doesn’t lie in any quadrant

Lies in quadrant II

Lies in quadrant IV

Quadrantal angle

Measuring Angles

Two usual ways of measuringDegrees

360± in one rotationRadians

2¼ radians in one rotation

Measuring Angles

Right angle: A quarter revolutionA right angle contains

90±

radians

¼2

Measuring Angles

Straight angle: A half revolution.A straight angle contains:

180±

¼ radians

Measuring Angles

Negative angles have negative measure

Multiple revolutions are allowed

Degrees, Minutes and Seconds

One complete revolution = 360±

One minute:One-sixtieth of a degreeOne minute is denoted 10

1± = 600

One second: One-sixtieth of a minuteOne second is denoted 100

10 = 6000

Degrees, Minutes and Seconds

Example. Convert to a decimal in degreesProblem: 64±3502700 Answer:

Example. Convert to the D±M0S00 formProblem: 73.582±

Answer:

Radians

Central angle: An angle whose vertex is at the center of a circleCentral angles subtend an arc

on the circle

Radians

One radian is the measure of an angle which subtends an arc with length equal to the radius of the circle

Radians

IMPORTANT!Radians are dimensionlessIf an angle appears with no

units, it must be assumed to be in radians

Arc Length

Theorem. [Arc Length]For a circle of radius r, a central

angle of µ radians subtends an arc whose length s is

s = rµ

WARNING!The angle must be given in

radians

Arc Length

Example. Problem: Find the length of the

arc of a circle of radius 5 centimeters subtended by a central angle of 1.4 radians

Answer:

Radians vs. Degrees

1 revolution = 2¼ radians = 360±

180± = ¼ radians1± = radians1 radian =

¼180

180¼

±

Radians vs. Degrees

Example. Convert each angle in degrees to radians and each angle in radians to degrees(a) Problem: 45±

Answer: (b) Problem: {270±

Answer: (c) Problem: 2 radians

Answer:

Radians vs. Degrees

Measurements of common angles

Area of a Sector of a Circle

Theorem. [Area of a Sector]The area A of the sector of a

circle of radius r formed by a central angle of µ radians is

A = 12r2µ

Area of a Sector of a Circle

Example.Problem: Find the area of the

sector of a circle of radius 3 meters formed by an angle of 45±. Round your answer to two decimal places.

Answer:

WARNING!The angle again must be given

in radians

Linear and Angular Speed

Object moving around a circle or radius r at a constant speedLinear speed: Distance traveled

divided by elapsed time

t = timeµ = central angle swept out in time ts = rµ = arc length = distance traveled

v = st


Object moving around a circle or radius r at a constant speedAngular speed: Angle swept out

divided by elapsed time

Linear and angular speeds are related

v = r!

! = µt


Example. A neighborhood carnival has a Ferris wheel whose radius is 50 feet. You measure the time it takes for one revolution to be 90 seconds.(a) Problem: What is the linear

speed (in feet per second) of this Ferris wheel?

Answer: (b) Problem: What is the angular

speed (in radians per second)? Answer:

Key Points

Basic Terminology Measuring Angles Degrees, Minutes and Seconds Radians Arc Length Radians vs. Degrees Area of a Sector of a Circle Linear and Angular Speed

Trigonometric Functions: Unit Circle Approach

Section 5.2

Unit Circle

Unit circle: Circle with radius 1 centered at the originEquation: x2 + y2 = 1Circumference: 2¼

Unit Circle

Travel t units around circle, starting from the point (1,0), ending at the point P = (x, y)

The point P = (x, y) is used to define the trigonometric functions of t


Let t be a real number and P = (x, y) the point on the unit circle corresponding to t:Sine function: y-coordinate of P

sin t = yCosine function: x-coordinate of

Pcos t = x

Tangent function: if x 0


Let t be a real number and P = (x, y) the point on the unit circle corresponding to t:Cosecant function: if y 0

Secant function: if x 0

Cotangent function: if y 0

Exact Values Using Points on the Circle

A point on the unit circle will satisfy the equation x2 + y2 = 1

Use this information together with the definitions of the trigonometric functions.

Exact Values Using Points on the Circle

Example. Let t be a real number and P = the point on the unit circle that corresponds to t.

Problem: Find the values of sin t, cos t, tan t, csc t, sec t and cot t

Answer:

Trigonometric Functions of Angles

Convert between arc length and angles on unit circle

Use angle µ to define trigonometric functions of the angle µ

Exact Values for Quadrantal Angles

Quadrantal angles correspond to integer multiples of 90± or of radians


Example. Find the values of the trigonometric functions of µProblem: µ = 0 = 0±

Answer:


Example. Find the values of the trigonometric functions of µProblem: µ = = 90±

Answer:


Example. Find the values of the trigonometric functions of µProblem: µ = ¼ = 180±

Answer:



Answer:


Example. Find the exact values of(a) Problem: sin({90±)

Answer: (b) Problem: cos(5¼)

Answer:

Exact Values for Standard Angles


Answer:



Answer:


Example. Find the values of the following expressions

(a) Problem: sin(315±)

Answer:

(b) Problem: cos({120±)

Answer:

(c) Problem:

Answer:

Approximating Values Using a Calculator

IMPORTANT!Be sure that your calculator is

in the correct mode.

Use the basic trigonometric facts:

Approximating Values Using a Calculator

Example. Use a calculator to find the approximate values of the following. Express your answers rounded to two decimal places.(a) Problem: sin 57±

Answer: (b) Problem: cot {153±

Answer: (c) Problem: sec 2

Answer:

Circles of Radius r

Theorem.For an angle µ in standard

position, let P = (x, y) be the point on the terminal side of µ that is also on the circle x2 + y2 = r2. Then

Circles of Radius r

Example. Problem: Find the exact values

of each of the trigonometric functions of an angle µ if ({12, {5) is a point on its terminal side.

Answer:

Key PointsUnit CircleTrigonometric FunctionsExact Values Using Points on the

Circle Trigonometric Functions of AnglesExact Values for Quadrantal

AnglesExact Values for Standard AnglesApproximating Values Using a

Calculator

Key Points (cont.)

Circles of Radius r

Properties of the Trigonometric Functions

Section 5.3

Domains of Trigonometric

FunctionsDomain of sine and cosine

functions is the set of all real numbers

Domain of tangent and secant functions is the set of all real numbers, except odd integer multiples of = 90±

Domain of cotangent and cosecant functions is the set of all real numbers, except integer multiples of ¼ = 180±

Ranges of Trigonometric

Functions Sine and cosine have range [{1,

1]{1 · sin µ · 1; jsin µj · 1{1 · cos µ · 1; jcos µj · 1

Range of cosecant and secant is ({1, {1] [ [1, 1) jcsc µj ¸ 1 jsec µj ¸ 1

Range of tangent and cotangent functions is the set of all real numbers

Periods of Trigonometric

FunctionsPeriodic function: A function

f with a positive number p such that whenever µ is in the domain of f, so is µ + p, and

f(µ + p) = f(µ) (Fundamental) period of f:

smallest such number p, if it exists


Functions Periodic Properties:

sin(µ + 2¼) = sin µcos(µ + 2¼) = cos µtan(µ + ¼) = tan µcsc(µ + 2¼) = csc µsec(µ + 2¼) = sec µcot(µ + ¼) = cot µ

Sine, cosine, cosecant and secant have period 2¼

Tangent and cotangent have period ¼


FunctionsExample. Find the exact values

of

(a) Problem: sin(7¼)

Answer:

(b) Problem:

Answer:

(c) Problem:

Answer:

Signs of the Trigonometric

FunctionsP = (x, y) corresponding to

angle µDefinitions of functions, where

defined

Find the signs of the functionsQuadrant I: x > 0, y > 0Quadrant II: x < 0, y > 0Quadrant III: x < 0, y < 0Quadrant IV: x > 0, y < 0


Functions


FunctionsExample:

Problem: If sin µ < 0 and cos µ > 0, name the quadrant in which the angle µ lies

Answer:

Quotient Identities

P = (x, y) corresponding to angle µ:

Get quotient identities:

Quotient Identities

Example.

Problem: Given and

, find the exact values of the

four remaining trigonometric

functions of µ using identities.

Answer:

Pythagorean Identities

Unit circle: x2 + y2 = 1 (sin µ)2 + (cos µ)2 = 1

sin2 µ + cos2 µ = 1tan2 µ + 1 = sec2 µ1 + cot2 µ = csc2 µ


Example. Find the exact values of each expression. Do not use a calculator

(a) Problem: cos 20± sec 20±

Answer:

(b) Problem: tan2 25± { sec2 25±

Answer:


Example.

Problem: Given that

and that µ is in Quadrant II,

find cos µ.

Answer:

Even-Odd Properties

A function f is even if f({µ) = f(µ) for all µ in the domain of f

A function f is odd if f({µ) = {f(µ) for all µ in the domain of f

Even-Odd PropertiesTheorem. [Even-Odd Properties]

sin({µ) = {sin(µ)cos({µ) = cos(µ)

tan({µ) = {tan(µ)csc({µ) = {csc(µ)sec({µ) = sec(µ)cot({µ) = {cot(µ)

Cosine and secant are even functions

The other functions are odd functions

Even-Odd Properties

Example. Find the exact values

of

(a) Problem: sin({30±)

Answer:

(b) Problem:

Answer:

(c) Problem:

Answer:

Fundamental Trigonometric IdentitiesQuotient Identities

Reciprocal Identities


Even-Odd Identities

Key Points Domains of Trigonometric Functions Ranges of Trigonometric Functions Periods of Trigonometric Functions Signs of the Trigonometric

Functions Quotient Identities Pythagorean Identities Even-Odd Properties Fundamental Trigonometric

Identities

Graphs of the Sine and Cosine Functions

Section 5.4

Graphing Trigonometric Functions

Graph in xy-plane Write functions as

y = f(x) = sin x y = f(x) = cos x y = f(x) = tan x y = f(x) = csc x y = f(x) = sec x y = f(x) = cot x

Variable x is an angle, measured in radians Can be any real number

Graphing the Sine Function

Periodicity: Only need to graph on interval [0, 2¼] (One cycle)

Plot points and graph

Properties of the Sine Function

Domain: All real numbers

Range: [{1, 1]

Odd function

Periodic, period 2¼

x-intercepts: …, {2¼, {¼, 0, ¼, 2¼, 3¼, …

y-intercept: 0

Maximum value: y = 1, occurring at

Minimum value: y = {1, occurring at

2

32

2 52

32

-4

-2

2

4

Transformations of the Graph of the Sine

FunctionsExample.

Problem: Use the graph of y =

sin x to graph

Answer:

Graphing the Cosine Function

Periodicity: Again, only need to graph on interval [0, 2¼] (One cycle)


Properties of the Cosine Function

Domain: All real numbers Range: [{1, 1] Even function Periodic, period 2¼ x-intercepts: y-intercept: 1 Maximum value: y = 1, occurring at

x = …, {2¼, 0, 2¼, 4¼, 6¼, … Minimum value: y = {1, occurring at

x = …, {¼, ¼, 3¼, 5¼, …

2

32

2 52

32

-4

-2

2

4

Example.


cos x to graph

Answer:

Transformations of the Graph of the Cosine

Functions

Sinusoidal Graphs

Graphs of sine and cosine functions appear to be translations of each other

Graphs are called sinusoidalConjecture.

Amplitude and Period of Sinusoidal Functions

Graphs of functions y = A sin x and y = A cos x will always satisfy inequality {jAj · y · jAjNumber jAj is the amplitude

Amplitude and Period of Sinusoidal

FunctionsGraphs of functions y = A sin

x and y = A cos x will always satisfy inequality {jAj · y · jAjNumber jAj is the amplitude

2

32

2 52

32

-4

-2

2

4

2

32

2 52

32

-4

-2

2

4


Period of y = sin(!x) and

y = cos(!x) is

2

32

2 52

32

-4

-2

2

4

2

32

2 52

32

-4

-2

2

4

Amplitude and Period of Sinusoidal

FunctionsCycle: One period of y =

sin(!x) or

y = cos(!x)

2

32

2 52

32

-4

-2

2

4

2

32

2 52

32

-4

-2

2

4


Cycle: One period of y =

sin(!x) or

y = cos(!x)


Theorem. If ! > 0, the

amplitude and period of y =

Asin(!x) and

y = Acos(! x) are given by

Amplitude = j Aj

Period = .


Example.Problem: Determine the

amplitude and period of y = {2cos(¼x)

Answer:

Graphing Sinusoidal Functions

One cycle contains four important subintervals

For y = sin x and y = cos x these are

Gives five key points on graph

2

3 2

2 5 2

32

-4

-2

2

4


Example.Problem: Graph y = {3cos(2x)Answer:

Finding Equations for Sinusoidal Graphs

Example.

Problem: Find an equation for

the graph.

Answer:

2

322 5

23

2

32

252

3

-6

-4

-2

2

4

6

Key Points

Graphing Trigonometric Functions

Graphing the Sine FunctionProperties of the Sine FunctionTransformations of the Graph of

the Sine FunctionsGraphing the Cosine FunctionProperties of the Cosine FunctionTransformations of the Graph of

the Cosine Functions

Key Points (cont.)

Sinusoidal GraphsAmplitude and Period of

Sinusoidal FunctionsGraphing Sinusoidal

FunctionsFinding Equations for

Sinusoidal Graphs

Graphs of the Tangent, Cotangent, Cosecant and Secant FunctionsSection 5.5

Graphing the Tangent Function

Periodicity: Only need to graph on interval [0, ¼]


Properties of the Tangent Function

Domain: All real numbers, except odd multiples of

Range: All real numbers Odd function Periodic, period ¼ x-intercepts: …, {2¼, {¼, 0, ¼, 2¼,

3¼, … y-intercept: 0 Asymptotes occur at

2

32

2 52

32

-8

-6

-4

-2

2

4

6

8

Transformations of the Graph of the Tangent

FunctionsExample.


tan x to graph

Answer:

Graphing the Cotangent Function

Periodicity: Only need to graph on interval [0, ¼]

Graphing the Cosecant and Secant Functions

Use reciprocal identitiesGraph of y = csc x

Graphing the Cosecant and Secant Functions

Use reciprocal identitiesGraph of y = sec x

Key Points

Graphing the Tangent FunctionProperties of the Tangent

FunctionTransformations of the Graph

of the Tangent FunctionsGraphing the Cotangent

FunctionGraphing the Cosecant and

Secant Functions

Phase Shifts; Sinusoidal Curve Fitting

Section 5.6


y = A sin(!x), ! > 0Amplitude jAjPeriod

y = A sin(!x { Á)Phase shiftPhase shift indicates amount of

shift To right if Á > 0

To left if Á < 0


Graphing y = A sin(!x { Á) or y = A cos(!x { Á):

Determine amplitude jAj

Determine period

Determine starting point of one

cycle:

Determine ending point of one

cycle:


Graphing y = A sin(!x { Á) or y = A cos(!x { Á):

Divide interval into

four subintervals, each with

length

Use endpoints of subintervals to

find the five key points

Fill in one cycle


Graphing y = A sin(!x { Á) or

y = A cos(!x { Á):

Extend the graph in each

direction to make it complete


Example. For the equation

(a) Problem: Find the amplitude

Answer:

(b) Problem: Find the period

Answer:

(c) Problem: Find the phase shift

Answer:

Finding a Sinusoidal Function from Data

Example. An experiment in a wind tunnel generates cyclic waves. The following data is collected for 52 seconds.Let v represent the wind speed in feet per second and let x represent the time in seconds.

Time (in seconds), x Wind speed (in feet per second), v

0 21

12 42

26 67

41 40

52 20


Example. (cont.)Problem: Write a sine equation

that represents the dataAnswer:

Key Points



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