SECTION 4 - crunchy math

Precalculus with Limits: A Graphing Approach 5e, Warm-Up Exercises and Daily Homework Quiz TransparenciesCopyright © Houghton Mifflin Company. All rights reserved.

34

SECTION 4.1

WARM-UP EXERCISES

Simplify.

1.

7π3

− 2π 2. −

5π6

+ 2π

Find the measure of the angle in degrees between the hour hand and the minutehand of a clock at the time shown. Measure the angle in the clockwise direction.

3. 12:30 4. 5:40

Find the area of the circle with the given radius.

5. r = 3 in. 6. r = 1.5 ft

DAILY HOMEWORK QUIZ

Determine the quadrant in which each angle lies. Then convert the angle measurefrom radians to degrees. Round to two decimal places.

1.

2π5

2. −1.8π 3.

7

3

Find the length of the arc on a circle of radius r intercepted by a central angle θ.Round to two decimal places.

4. r = 11 in.; θ = 100º 5. r = 3.6 ft; θ = 290º 6. r = 20 cm; θ =

π3

ANSWERS

WU 1.

π3

2.

7π6

3. 165º 4. 70º

5. 9π in.2 6. 2.25π ft2

QUIZ 1. Quadrant I; 72º 2. Quadrant I; –324º

3. Quadrant II; 133.69º 4. 19.20 in.

5. 18.22 ft 6. 20.94 cm


35

SECTION 4.2

WARM-UP EXERCISES

Rewrite each angle in degree measure. (Do not use a calculator.)

1.

3π4

2.

10π3

3. −

13π6

4.

π12

Classify each function as odd, even, or neither.

5. f(x) = 6x3 + 3x 6. g(x) = |x| 7. f(x) = 2x2 – 8x

DAILY HOMEWORK QUIZ

Evaluate (if possible) the six trigonometric functions of the real number.

1. t =

3π4

2. t = −

11π6

3. t =

π3

Use the value of the trigonometric function to evaluate the given functions.

4. sin t = 0.8 (a) sin (–t) (b) csc t

5. cos (–t) = − 3

5(a) cos (! – t) (b) sec t


36

ANSWERS

WU 1. 135º 2. 600º 3. –30° 4. 15º

5. odd 6. even 7. neither

QUIZ 1. sin

3π4

⎛⎝⎜

⎞⎠⎟=

2

2; cos

3π4

⎛⎝⎜

⎞⎠⎟= −

2

2; tan

3π4

⎛⎝⎜

⎞⎠⎟= −1;

csc

3π4

⎛⎝⎜

⎞⎠⎟= 2; sec

3π4

⎛⎝⎜

⎞⎠⎟= − 2; cot

3π4

⎛⎝⎜

⎞⎠⎟= −1

2. sin −

11π6

⎛⎝⎜

⎞⎠⎟=

1

2; cos −

11π6

⎛⎝⎜

⎞⎠⎟=

3

2; tan −

11π6

⎛⎝⎜

⎞⎠⎟=

3

3;

csc −

11π6

⎛⎝⎜

⎞⎠⎟= 2; sec −

11π6

⎛⎝⎜

⎞⎠⎟=

2 3

3; cot −

11π6

⎛⎝⎜

⎞⎠⎟= 3

3. sin

π3

⎛⎝⎜

⎞⎠⎟=

3

2; cos

π3

⎛⎝⎜

⎞⎠⎟=

1

2; tan

π3

⎛⎝⎜

⎞⎠⎟= 3;

csc

π3

⎛⎝⎜

⎞⎠⎟=

2 3

3; sec

π3

⎛⎝⎜

⎞⎠⎟= 2; cot

π3

⎛⎝⎜

⎞⎠⎟=

3

3

4. (a) –0.8 (b) 1.25

5. (a)

3

5(b)

−

5

3


37

SECTION 4.3

WARM-UP EXERCISES

The lengths of the legs of a right triangle are given. Find the length of the hypotenuse.

1. 3 cm, 5 cm

2. 2 in., 6 in.

Evaluate (if possible) the six trigonometric functions of the real number.

3. t =

5π4

4. t = −

3π2

5. t =

π6

DAILY HOMEWORK QUIZ

1. Find the exact values of the sixtrigonometric functions of the angleθ for the triangle.

Use trigonometric identities to transform the left side of the equation into the right side(0 < θ < !/2).

2. tan2α csc2α = sec2α3. sinθ + cotθ cosθ = cscθ


38

ANSWERS

WU 1. 34 cm 2. 2 10 in.

3. sin

5π4

⎛⎝⎜

⎞⎠⎟= −

2

2; cos

5π4

⎛⎝⎜

⎞⎠⎟= −

2

2; tan

5π4

⎛⎝⎜

⎞⎠⎟= 1;

csc

5π4

⎛⎝⎜

⎞⎠⎟= − 2; sec

5π4

⎛⎝⎜

⎞⎠⎟= − 2; cot

5π4

⎛⎝⎜

⎞⎠⎟= 1

4. sin −

3π2

⎛⎝⎜

⎞⎠⎟= 1; cos −

3π2

⎛⎝⎜

⎞⎠⎟= 0; csc −

3π2

⎛⎝⎜

⎞⎠⎟= 1; cot −

3π2

⎛⎝⎜

⎞⎠⎟= 0

5. sin

π6

⎛⎝⎜

⎞⎠⎟=

1

2; cos

π6

⎛⎝⎜

⎞⎠⎟=

3

2; tan

π6

⎛⎝⎜

⎞⎠⎟=

3

3;

csc

π6

⎛⎝⎜

⎞⎠⎟= 2; sec

π6

⎛⎝⎜

⎞⎠⎟=

2 3

3; cot

π6

⎛⎝⎜

⎞⎠⎟= 3

QUIZ 1. sinθ =

3

7; cosθ =

2 10

7; tanθ =

3 10

20;

cscθ =

7

3; secθ =

7 10

20; cotθ =

2 10

3

2. tan2α csc2α =

sin2αcos2α

⎛

⎝⎜⎞

⎠⎟1

sin2α⎛⎝⎜

⎞⎠⎟=

1

cos2α= sec2α

3. sinθ + cotθ cosθ = sinθ +

cosθsinθ

(cosθ ) = sinθ +cos2θsinθ

=

sin2θsinθ

+cos2θsinθ

=sin2θ + cos2θ

sinθ=

1

sinθ= cscθ


39

SECTION 4.4

WARM-UP EXERCISES

Determine the quadrant in which each angle lies.

1. 77º 2. 240º 3. –9º

Find the values of θ in degrees (0º < θ < 90º) and radians (0 < θ < !/2) without theaid of a calculator.

4. cosθ =

3

25.

tanθ =

3

36. cscθ = 2

DAILY HOMEWORK QUIZ

Find the values of the six trigonometric functions of θ with the given constraint.

1. tan θ = 2; θ lies in Quadrant III.

2. sin θ = −

1

6; cos θ > 0

3. sec θ is undefined; ! < θ < 2!

Evaluate the sine, cosine, and tangent of the angle without using a calculator.

4. 240º

5. −

9π4


40

ANSWERS

WU 1. Quadrant I 2. Quadrant III 3. Quadrant IV

4. 30º,

π6

5. 30º,

π6

6. 45º,

π4

QUIZ 1. sinθ = −

2 3

3; cosθ = −

3

3; tanθ = 2;

cscθ = −

3

2; secθ = − 3; cotθ =

1

2

2. sinθ = −

1

6; cosθ = −

35

6; tanθ = −

35

35;

cscθ = −6; secθ = −

6 35

35; cotθ = − 35

3. sinθ = −1; cosθ = 0; cscθ = −1; cotθ = 0

4. sin 240° = −

3

2; cos 240° = −

1

2; tan 240° = 3

5. sin −

9π4

⎛⎝⎜

⎞⎠⎟= −

2

2; cos −

9π4

⎛⎝⎜

⎞⎠⎟=

2

2; tan −

9π4

⎛⎝⎜

⎞⎠⎟= −1


41

SECTION 4.5

WARM-UP EXERCISES

Evaluate the trigonometric function of the quadrant angle.

1. cos

π2

⎛⎝⎜

⎞⎠⎟

2. sin

3π2

⎛⎝⎜

⎞⎠⎟

Find two solutions of the equation. Give your answers in radians (0 < θ < 2!).

3. cos(θ ) = −

1

2

4. sin(θ ) =

3

2

5. cos(θ ) = 0

DAILY HOMEWORK QUIZ

Find the period and amplitude.

1. y = −3cosπx

2. y =

3

2sin

x

4

Sketch the graph of the function. (Include two full periods.)

3. y = −cos(x −π )

4. y =

2

3sin

πx

3

⎛⎝⎜

⎞⎠⎟+ 2

5. y = 1− cos(2x )


42

ANSWERS

WU 1. 0 2. –1 3.

2π3

, 4π3

4.

π3

, 2π3

5.

π2

, 3π2

QUIZ 1. 3; 2 2.

3

2; π2

3.

4.

5.


43

SECTION 4.6

WARM-UP EXERCISES


1. y = −2sin(πx )

2. y = 2+ cos(x −π )

3. y =

1

2sin

x

2−π4

⎛⎝⎜

⎞⎠⎟

Find the period and the amplitude of the trigonometric function.

4. y = 5cos

πx

3

⎛⎝⎜

⎞⎠⎟

5. y = −sin(3x +π )

6. y =

2

3cos(x )

DAILY HOMEWORK QUIZ


1. y = 2tanπx −1

2. y =

1

2sec x +

π2

⎛⎝⎜

⎞⎠⎟

3. y = −2+ cot(2πx )

Use a graph to solve the equation on the interval [–2!, 2!].

4. cot x = −1

5. sec x =

2 3

3

6. csc x = −2


44

ANSWERS

WU 1. 2.

3.

4. 6; 5 5.

2π3

; 1 6. 2π ;

2

3

QUIZ 1. 2.


45

3.

4. −

5π4

, −π4

, 3π4

, 7π4

5. −

11π6

, −π6

, π6

, 11π

6

6. −

5π6

, −π6

, 7π6

, 11π

6


46

SECTION 4.7

WARM-UP EXERCISES

Use the properties of inverse functions to evaluate the expression.

1. f–1(f(–2)) 2. f(f–1(13))

Expressions representing the legs of a right triangle are given. Write an expression torepresent the hypotenuse.

3. 5x; 3 4. (x – 2); x

Use a calculator to evaluate the expression. Round your result to two decimal places.

5. sin 1.9 6. csc (–2) 7. cot (11º)

DAILY HOMEWORK QUIZ

Evaluate the expression without using a calculator.

1. arccos 1 2.

tan−1 3

3

⎛

⎝⎜

⎞

⎠⎟

Find the exact value of the expression. (Hint: Sketch a right triangle.)

3. tan arccos −

4

5

⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥

4. csc tan−1(3)⎡⎣ ⎤⎦

Write an algebraic expression that is equivalent to the expression.

5. sin (arccos 2x) 6. cot arcsin

2

x

⎛⎝⎜

⎞⎠⎟

ANSWERS

WU 1. –2 2. 13 3. 25x 2 + 9 4. 2x 2 − 4x + 4

5. 0.95 6. –1.10 7. 5.14

QUIZ 1. 0 2.

π6

3. −

3

44.

10

3

5. 1− 4x 2 6.

x 2 − 4

2


47

SECTION 4.8

WARM-UP EXERCISES

Use a calculator to evaluate the expression. Round your result to two decimal places.

1. arccos 0.73

2. arctan 3

3. sin–1 0.45

4. tan–1 (–1.5)

5. A child holds a balloon in her hand at a height of 3 feet. The balloon is attached toa 25-inch string, and the wind blows the balloon so the string forms an angle θfrom vertical. Write θ as a function of the height h of the balloon, in inches.

DAILY HOMEWORK QUIZ

1. The sun is 13º above the horizon. Find the height of a road sign that casts ashadow 18 feet long. Round your answer to two decimal places.

2. An airplane is 120 miles south and 45 miles east of an airport. The pilot wants tofly directly to the airport. What bearing should be taken? Round your answer totwo decimal places.

3. A piece of paper measures 8 inches by 10 inches. A corner is folded over so that ittouches the long side at its midpoint. What is the angle formed by the corner andthe long side? Round your answer to the nearest degree.

ANSWERS

WU 1. 0.75 2. 1.25 3. 0.47

4. –0.98 5. θ = cos−1 h − 3

25

⎛⎝⎜

⎞⎠⎟

QUIZ 1. 4.16 ft 2. N 22.02° E 3. 26°

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