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© 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 2 of 39
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
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 3 of 39
§ 8.1
Radian Measure of Angles
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 4 of 39
Radians and Degrees
Positive and Negative Angles
Converting Degrees to Radians
Determining an Angle
Section Outline
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 5 of 39
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.
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 6 of 39
Radians and Degrees
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 7 of 39
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
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 8 of 39
Converting Degrees to Radians
EXAMPLEEXAMPLE
SOLUTIONSOLUTION
Convert the following to radian measure .210 450 ba
2
5radians
180450450
a
6
7radians
180210210
b
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 9 of 39
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
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 10 of 39
§ 8.2
The Sine and the Cosine
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 11 of 39
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
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 12 of 39
Sine & Cosine
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 13 of 39
Sine & Cosine in a Right Triangle
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 14 of 39
Sine & Cosine in a Unit Circle
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 15 of 39
Properties of Sine & Cosine
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 16 of 39
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
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 17 of 39
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.
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 18 of 39
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
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 19 of 39
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
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 20 of 39
The Graphs of Sine & Cosine
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 21 of 39
§ 8.3
Differentiation and Integration of sin t and cos t
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 22 of 39
Derivatives of Sine and Cosine
Differentiating Sine and Cosine
Differentiating Cosine in Application
Application of Differentiating and Integrating Sine
Section Outline
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 23 of 39
Derivatives of Sine & Cosine
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 24 of 39
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
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 25 of 39
Differentiating Cosine in Application
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.
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 26 of 39
Differentiating Cosine in Application
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.
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 27 of 39
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?
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 28 of 39
Application of Differentiating & Integrating Sine
CONTINUECONTINUEDD
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 29 of 39
Application of Differentiating & Integrating Sine
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
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 30 of 39
Application of Differentiating & Integrating Sine
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.
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 31 of 39
§ 8.4
The Tangent and Other Trigonometric Functions
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 32 of 39
Other Trigonometric Functions
Other Trigonometric Identities
Applications of Tangent
Derivative Rules for Tangent
Differentiating Tangent
The Graph of Tangent
Section Outline
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 33 of 39
Other Trigonometric Functions
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 34 of 39
Other Trigonometric Identities
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 35 of 39
Applications of Tangent
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
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 36 of 39
Applications of Tangent
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
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 37 of 39
Derivative Rules for Tangent
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 38 of 39
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
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 39 of 39
The Graph of Tangent