© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 1 of 39 Chapter 8 The Trigonometric Functions.

© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 1 of 39

Chapter 8

The Trigonometric Functions


Radian Measure of Angles

The Sine and the Cosine

Differentiation and Integration of sin t and cos t

The Tangent and Other Trigonometric Functions

Chapter Outline


§ 8.1

Radian Measure of Angles


Radians and Degrees

Positive and Negative Angles

Converting Degrees to Radians

Determining an Angle

Section Outline


Radians and Degrees

The central angle determined by an arc of length 1 along the circumference of a circle is said to have a

measure of one radian.


Radians and Degrees


Positive & Negative Angles

Definition Example

Positive Angle: An angle measured in the counter-clockwise direction

Definition Example

Negative Angle: An angle measured in the clockwise direction


Converting Degrees to Radians

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

Convert the following to radian measure .210 450 ba

2

5radians

180450450

a

6

7radians

180210210

b


Determining an Angle

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

Give the radian measure of the angle described.

The angle above consists of one full revolution (2π radians) plus one half-revolutions (π radians). Also, the angle is clockwise and therefore negative. That is,

.32 t


§ 8.2

The Sine and the Cosine


Sine and Cosine

Sine and Cosine in a Right Triangle

Sine and Cosine in a Unit Circle

Properties of Sine and Cosine

Calculating Sine and Cosine

Using Sine and Cosine

Determining an Angle t

The Graphs of Sine and Cosine

Section Outline


Sine & Cosine


Sine & Cosine in a Right Triangle


Sine & Cosine in a Unit Circle


Properties of Sine & Cosine


Calculating Sine & Cosine

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

Give the values of sin t and cos t, where t is the radian measure of the angle shown.

Since we wish to know the sine and cosine of the angle that measures t radians, and because we know the length of the side opposite the angle as well as the hypotenuse, we can immediately determine sin t.

4

1sin t

Since sin2t + cos2t = 1, we have


Calculating Sine & Cosine

Replace sin2t with (1/4)2.1cos4

1 22

t

CONTINUECONTINUEDD

1cos16

1 2 t Simplify.

16

15cos2 t Subtract.

4

15cos t Take the square root of both

sides.


Using Sine & Cosine

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

If t = 0.4 and a = 10, find c.

Since cos(0.4) = 10/c, we get

c

104.0cos

104.0cos c

.9.104.0cos

10c


Determining an Angle t

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

Find t such that –π/2 ≤ t ≤ π/2 and t satisfies the stated condition.

One of our properties of sine is sin(-t) = -sin(t). And since -sin(3π/8) = sin(-3π/8) and –π/2 ≤ -3π/8 ≤ π/2, we have t = -3π/8.

8/3sinsin t


The Graphs of Sine & Cosine


§ 8.3

Differentiation and Integration of sin t and cos t


Derivatives of Sine and Cosine

Differentiating Sine and Cosine

Differentiating Cosine in Application

Application of Differentiating and Integrating Sine

Section Outline


Derivatives of Sine & Cosine


Differentiating Sine & Cosine

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

Differentiate the following.

3cos sin b a πte x

xexdx

dee

dx

d xxx sincos a coscoscos

tdt

dtt

dt

dπt

dt

d sinsin3

1sinsin b 32313

tdt

dtt cossin

3

1 32

tt cossin3

1 32



EXAMPLEEXAMPLE

SOLUTIOSOLUTIONN

Suppose that a person’s blood pressure P at time t (in seconds) is given by P = 100 + 20cos 6t.

tP 6cos20100

Find the maximum value of P (called the systolic pressure) and the minimum value of P (called the diastolic pressure) and give one or two values of t where these maximum and minimum values of P occur.

The maximum value of P and the minimum value of P will occur where the function has relative minima and maxima. These relative extrema occur where the value of the first derivative is zero.

This is the given function.

ttP 6sin12066sin20 Differentiate.

06sin120 t Set P΄ equal to 0.

06sin t Divide by -120.



Notice that sin6t = 0 when 6t = 0, π, 2π, 3π,... That is, when t = 0, π/6, π/3, π/2,... Now we can evaluate the original function at these values for t.

CONTINUECONTINUEDD

t 100 + 20cos6t

0 120

π/6 80

π/3 120

π/2 80

Notice that the values of the function P cycle between 120 and 80. Therefore, the maximum value of the function is 120 and the minimum value is 80.


Application of Differentiating & Integrating Sine

EXAMPLEEXAMPLE

(Average Temperature) The average weekly temperature in Washington, D.C. t weeks after the beginning of the year is

.1252

2sin2354

ttf

The graph of this function is sketched below.

(a) What is the average weekly temperature at week 18?

(b) At week 20, how fast is the temperature changing?



CONTINUECONTINUEDD



18

0

18

012

52

2sin2354

18

1

018

1dttdttf

(a) The time interval up to week 18 corresponds to t = 0 to t = 18. The average value of f (t) over this interval is

CONTINUECONTINUEDD

SOLUTIOSOLUTIONN

18

0

1252

2cos

2

522354

18

1

tt

13

6cos

5980

18

1

13

3cos

598972

18

1

.359.47944.2218

1521.829

18

1



Therefore, the average value of f (t) is about 47.359 degrees.

CONTINUECONTINUEDD

(b) To determine how fast the temperature is changing at week 20, we need to evaluate f ΄(20).

12

52

2sin2354 ttf

This is the given function.

52

212

52

2cos23

ttf Differentiate.

1226

cos26

23ttf

Simplify.

579.1122026

cos26

2320

f Evaluate f ΄(20).

Therefore, the temperature is changing at a rate of 1.579 degrees per week.


§ 8.4

The Tangent and Other Trigonometric Functions


Other Trigonometric Functions

Other Trigonometric Identities

Applications of Tangent

Derivative Rules for Tangent

Differentiating Tangent

The Graph of Tangent

Section Outline


Other Trigonometric Functions


Other Trigonometric Identities



EXAMPLEEXAMPLE

SOLUTIOSOLUTIONN

Find the width of a river at points A and B if the angle BAC is 90°, the angle ACB is 40°, and the distance from A to C is 75 feet.

7540tan

r

Let r denote the width of the river. Then equation (3) implies that

.40tan75 r

r



We convert 40° into radians. We find that 40° = (π/180)40 radians ≈ 0.7 radians, and tan(0.7) ≈ 0.84229. Hence

meters. 17.6384229.07540tan75 r

CONTINUECONTINUEDD


Derivative Rules for Tangent


Differentiating Tangent

EXAMPLEEXAMPLE

SOLUTIOSOLUTIONN

Differentiate.4tan2 2 xy

From equation (5) we find that

4tan2 2 xdx

d

dx

dyy

dx

d

44sec2 222 xdx

dx

442

14sec2 221222

x

dx

dxx

xxx 242

14sec2

21222

.

4

4sec22

22

x

xx


The Graph of Tangent

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 1 of 39 Chapter 8 The Trigonometric Functions.

Documents

pearson education

radian measure slide

cosine sine

unit circle slide

clockwise direction

right triangle slide

trigonometric functions

cosine section