Question 1

Name the quadrant in which the angle θ lies.
-$\sin \theta > 0 , \quad \cos \theta < 0$

Accepted Answer

A)  I 
B)  II 
C)  III 
D)  IV 
A)  I 
B)  II 
C)  III 
D)  IV

Question 2

Solve the problem.
-An irrigation sprinkler in a field of lettuce sprays water over a distance of 35 feet as it rotates through an angle of $135 ^ { \circ }$. What area of the field receives water? If necessary, round the answer to two decimal places.

Accepted Answer

A)  $459.38 \mathrm { ft } ^ { 2 }$ 
B)  $2,886.34 \mathrm { ft } ^ { 2 }$ 
C)  $1,443.17 \mathrm { ft } ^ { 2 }$ 
D)  $41.23 \mathrm { ft } ^ { 2 }$ 
A)  $459.38 \mathrm { ft } ^ { 2 }$ 
B)  $2,886.34 \mathrm { ft } ^ { 2 }$ 
C)  $1,443.17 \mathrm { ft } ^ { 2 }$ 
D)  $41.23 \mathrm { ft } ^ { 2 }$

Question 3

Find the exact value. Do not use a calculator.
-$$\csc \left( - \frac { \pi } { 2 } \right)$$

Accepted Answer

A)  1 
B)  $- 1$ 
C)  0 
D)  undefined 
A)  1 
B)  $- 1$ 
C)  0 
D)  undefined

Question 4

Solve the problem.
-For what numbers $\theta$ is $f ( \theta ) = \cot \theta$ not defined?

Accepted Answer

A)  integral multiples of $\pi \left( 180 ^ { \circ } \right)$ 
B)  all real numbers 
C)  odd multiples of $\pi \left( 180 ^ { \circ } \right)$ 
D)  odd multiples of $\frac { \pi } { 2 } \left( 90 ^ { \circ } \right)$ 
A)  integral multiples of $\pi \left( 180 ^ { \circ } \right)$ 
B)  all real numbers 
C)  odd multiples of $\pi \left( 180 ^ { \circ } \right)$ 
D)  odd multiples of $\frac { \pi } { 2 } \left( 90 ^ { \circ } \right)$

Question 5

Graph the sinusoidal function.
-$y=-3 \cos (\pi x)$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 6

Match the function to its graph.
-$$y = - \tan \left( x - \frac { \pi } { 2 } \right)$$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 7

Find the exact value of the expression if θ = 45°. Do not use a calculator.
-$f ( \theta ) = \sin \theta \quad$ Find $3 f ( \theta )$.

Accepted Answer

A)  $\frac { 3 \sqrt { 2 } } { 2 }$ 
B)  $- \frac { \sqrt { 2 } } { 2 }$ 
C)  $- \frac { 3 \sqrt { 2 } } { 2 }$ 
D)  $\frac { \sqrt { 2 } } { 2 }$ 
A)  $\frac { 3 \sqrt { 2 } } { 2 }$ 
B)  $- \frac { \sqrt { 2 } } { 2 }$ 
C)  $- \frac { 3 \sqrt { 2 } } { 2 }$ 
D)  $\frac { \sqrt { 2 } } { 2 }$

Question 8

Write the word or phrase that best completes each statement or answers the question.
-The strength S of a wooden beam with rectangular cross section is given by the formula $\mathrm { S } = \mathrm { kd } ^ { 3 } \sin ^ { 2 } \theta \cos \theta$ where d is the diagonal length, θ the angle illustrated, and k is a constant that varies with the type of wood used.   Let d = 1 and express the strength S in terms of the constant k for θ = 45°, 50°, 55°, 60°, and 65°. Does the strength
always increase as θ gets larger?

Accepted Answer

0.354k; 0.377k; 0.38

Question 9

Choose the one alternative that best completes the statement or answers the question.
Use the even-odd properties to find the exact value of the expression. Do not use a calculator.
-$\sin \left( - \frac { \pi } { 4 } \right)$

Accepted Answer

A)  $\frac { \sqrt { 3 } } { 2 }$ 
B)  $- \frac { \sqrt { 3 } } { 2 }$ 
C)  $- \frac { \sqrt { 2 } } { 2 }$ 
D)  $\frac { \sqrt { 2 } } { 2 }$ 
A)  $\frac { \sqrt { 3 } } { 2 }$ 
B)  $- \frac { \sqrt { 3 } } { 2 }$ 
C)  $- \frac { \sqrt { 2 } } { 2 }$ 
D)  $\frac { \sqrt { 2 } } { 2 }$

Question 10

Use the fact that the trigonometric functions are periodic to find the exact value of the expression. Do not use a calculator.
-$\tan 540 ^ { \circ }$

Accepted Answer

A)  $\frac { \sqrt { 3 } } { 3 }$ 
B)  0 
C)  $- 1$ 
D)  undefined 
A)  $\frac { \sqrt { 3 } } { 3 }$ 
B)  0 
C)  $- 1$ 
D)  undefined

Question 11

Find the area A. Round the answer to three decimal places.
-

Accepted Answer

A)  $1.963 \mathrm {~m} ^ { 2 }$ 
B)  $9.817 \mathrm {~m} ^ { 2 }$ 
C)  $6.25 \mathrm {~m} ^ { 2 }$ 
D)  $19.635 \mathrm {~m} ^ { 2 }$ 
A)  $1.963 \mathrm {~m} ^ { 2 }$ 
B)  $9.817 \mathrm {~m} ^ { 2 }$ 
C)  $6.25 \mathrm {~m} ^ { 2 }$ 
D)  $19.635 \mathrm {~m} ^ { 2 }$

Question 12

Choose the one alternative that best completes the statement or answers the question.
Use transformations to graph the function.
-$y=2 \sin (\pi-x)$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 13

Draw the angle.
-$\frac { 5 \pi } { 3 }$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 14

Draw the angle.
-$$- \frac { 7 \pi } { 6 }$$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 15

Use the fact that the trigonometric functions are periodic to find the exact value of the expression. Do not use a calculator.
-$$\cos \frac { 16 \pi } { 3 }$$

Accepted Answer

A)  $- \frac { \sqrt { 3 } } { 2 }$ 
B)  $\frac { \sqrt { 3 } } { 2 }$ 
C)  $\frac { 1 } { 2 }$ 
D)  $- \frac { 1 } { 2 }$ 
A)  $- \frac { \sqrt { 3 } } { 2 }$ 
B)  $\frac { \sqrt { 3 } } { 2 }$ 
C)  $\frac { 1 } { 2 }$ 
D)  $- \frac { 1 } { 2 }$

Question 16

Graph the function.
-$y=\csc \left(\frac{\pi x}{5}+\frac{3 \pi}{5}\right)$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 17

Solve the problem.
-If friction is ignored, the time t (in seconds) required for a block to slide down an inclined plane is given by the formula $t = \sqrt { \frac { 2 a } { g \sin \theta \cos \theta } }$ where a is the length (in feet) of the base and g =32 feet per second per second is the acceleration of gravity. How long does it take a block to slide down an inclined plane with base a = 12 when θ = 58°? If necessary, round the
Answer to the nearest tenth of a second.

Accepted Answer

A)  1.5 sec 
B)  1.4 sec 
C)  0.5 sec 
D)  1.6 sec 
A)  1.5 sec 
B)  1.4 sec 
C)  0.5 sec 
D)  1.6 sec

Question 18

Use the fact that the trigonometric functions are periodic to find the exact value of the expression. Do not use a calculator.
-$\sec \frac { 19 \pi } { 4 }$

Accepted Answer

A)  $- \sqrt { 2 }$ 
B)  $- \frac { 2 \sqrt { 3 } } { 3 }$ 
C)  $\frac { \sqrt { 2 } } { 2 }$ 
D)  $- 2$ 
A)  $- \sqrt { 2 }$ 
B)  $- \frac { 2 \sqrt { 3 } } { 3 }$ 
C)  $\frac { \sqrt { 2 } } { 2 }$ 
D)  $- 2$

Question 19

Solve the problem.
-For what numbers $x , 0 \leq x \leq 2 \pi$, does $\sin x = - 1$ ?

Accepted Answer

A)  $\pi$ 
B)  $\frac { \pi } { 2 } , \frac { 3 \pi } { 2 }$ 
C)  $\frac { 3 \pi } { 2 }$ 
D)  none 
A)  $\pi$ 
B)  $\frac { \pi } { 2 } , \frac { 3 \pi } { 2 }$ 
C)  $\frac { 3 \pi } { 2 }$ 
D)  none

Question 20

Solve the problem.
-The number of hours of sunlight in a day can be modeled by a sinusoidal function. In the northern hemisphere, the longest day of the year occurs at the summer solstice and the shortest day occurs at the winter solstice. In 2000, these
Dates were June 22 (the 172nd day of the year) and December 21 (the 356th day of the year), respectively. A town experiences $11.23$ hours of sunlight at the summer solstice and $8.03$ hours of sunlight at the winter solstice. Find a sinusoidal function $y = A \sin ( \omega x - \varphi ) + B$ that fits the data, where $x$ is the day of the year. (Note: There are 366 days in the year $2000 . )$

Accepted Answer

A)  $y = 1.6 \sin \left( \frac { \pi } { 183 } x - \frac { 161 \pi } { 366 } \right) + 9.63$ 
B)  $y = 11.23 \sin \left( \frac { 172 \pi } { 356 } x - \frac { 2 \pi } { 3 } \right) + 9.63$ 
C)  $y = 1.6 \sin \left( \frac { \pi } { 183 } x - \frac { 2 \pi } { 3 } \right) + 9.63$ 
D)  $y = 11.23 \sin \left( \pi x - \frac { 2 \pi } { 3 } \right) + 8.03$ 
A)  $y = 1.6 \sin \left( \frac { \pi } { 183 } x - \frac { 161 \pi } { 366 } \right) + 9.63$ 
B)  $y = 11.23 \sin \left( \frac { 172 \pi } { 356 } x - \frac { 2 \pi } { 3 } \right) + 9.63$ 
C)  $y = 1.6 \sin \left( \frac { \pi } { 183 } x - \frac { 2 \pi } { 3 } \right) + 9.63$ 
D)  $y = 11.23 \sin \left( \pi x - \frac { 2 \pi } { 3 } \right) + 8.03$

Name the quadrant in which the angle θ lies. - $\sin \theta > 0 , \quad \cos \theta < 0$

Solve the problem. -An irrigation sprinkler in a field of lettuce sprays water over a distance of 35 feet as it rotates through an angle of $135 ^ { \circ }$ . What area of the field receives water? If necessary, round the answer to two decimal places.

Find the exact value. Do not use a calculator. - $\csc \left( - \frac { \pi } { 2 } \right)$

Solve the problem. -For what numbers $\theta$ is $f ( \theta ) = \cot \theta$ not defined?

Graph the sinusoidal function. - $y=-3 \cos (\pi x)$ $Graph the sinusoidal function. - y=-3 \cos (\pi x)$

Match the function to its graph. - $y = - \tan \left( x - \frac { \pi } { 2 } \right)$

Find the exact value of the expression if θ = 45°. Do not use a calculator. - $f ( \theta ) = \sin \theta \quad$ Find $3 f ( \theta )$ .

Choose the one alternative that best completes the statement or answers the question. Use the even-odd properties to find the exact value of the expression. Do not use a calculator. - $\sin \left( - \frac { \pi } { 4 } \right)$

Use the fact that the trigonometric functions are periodic to find the exact value of the expression. Do not use a calculator. - $\tan 540 ^ { \circ }$

Find the area A. Round the answer to three decimal places. -

Choose the one alternative that best completes the statement or answers the question. Use transformations to graph the function. - $y=2 \sin (\pi-x)$ $Choose the one alternative that best completes the statement or answers the question. Use transformations to graph the function. - y=2 \sin (\pi-x)$

Draw the angle. - $\frac { 5 \pi } { 3 }$

Draw the angle. - $- \frac { 7 \pi } { 6 }$

Use the fact that the trigonometric functions are periodic to find the exact value of the expression. Do not use a calculator. - $\cos \frac { 16 \pi } { 3 }$

Graph the function. - $y=\csc \left(\frac{\pi x}{5}+\frac{3 \pi}{5}\right)$ $Graph the function. - y=\csc \left(\frac{\pi x}{5}+\frac{3 \pi}{5}\right)$

Use the fact that the trigonometric functions are periodic to find the exact value of the expression. Do not use a calculator. - $\sec \frac { 19 \pi } { 4 }$

Solve the problem. -For what numbers $x , 0 \leq x \leq 2 \pi$ , does $\sin x = - 1$ ?

Functions and Their Graphs

Linear and Quadratic Functions

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Analytic Trigonometry

Applications of Trigonometric Functions

Polar Coordinates; Vectors

Analytic Geometry

Systems of Equations and Inequalities

Sequences; Induction; the Binomial Theorem

Counting and Probability

A Preview of Calculus: the Limit, Derivative, and Integral of a Function

Foundations: a Prelude to Functions

Graphing Utilities

Filters

Exam 5: Trigonometric Functions

Name the quadrant in which the angle θ lies. - sin⁡θ>0,cos⁡θ<0\sin \theta > 0 , \quad \cos \theta < 0sinθ>0,cosθ<0

Solve the problem. -An irrigation sprinkler in a field of lettuce sprays water over a distance of 35 feet as it rotates through an angle of 135∘135 ^ { \circ }135∘ . What area of the field receives water? If necessary, round the answer to two decimal places.

Find the exact value. Do not use a calculator. - csc⁡(−π2)\csc \left( - \frac { \pi } { 2 } \right)csc(−2π​)

Solve the problem. -For what numbers θ\thetaθ is f(θ)=cot⁡θf ( \theta ) = \cot \thetaf(θ)=cotθ not defined?

Graph the sinusoidal function. - y=−3cos⁡(πx)y=-3 \cos (\pi x)y=−3cos(πx)

Match the function to its graph. - y=−tan⁡(x−π2)y = - \tan \left( x - \frac { \pi } { 2 } \right)y=−tan(x−2π​)

Find the exact value of the expression if θ = 45°. Do not use a calculator. - f(θ)=sin⁡θf ( \theta ) = \sin \theta \quadf(θ)=sinθ Find 3f(θ)3 f ( \theta )3f(θ) .

Choose the one alternative that best completes the statement or answers the question. Use the even-odd properties to find the exact value of the expression. Do not use a calculator. - sin⁡(−π4)\sin \left( - \frac { \pi } { 4 } \right)sin(−4π​)

Use the fact that the trigonometric functions are periodic to find the exact value of the expression. Do not use a calculator. - tan⁡540∘\tan 540 ^ { \circ }tan540∘

Find the area A. Round the answer to three decimal places. -

Choose the one alternative that best completes the statement or answers the question. Use transformations to graph the function. - y=2sin⁡(π−x)y=2 \sin (\pi-x)y=2sin(π−x)

Draw the angle. - 5π3\frac { 5 \pi } { 3 }35π​

Draw the angle. - −7π6- \frac { 7 \pi } { 6 }−67π​

Use the fact that the trigonometric functions are periodic to find the exact value of the expression. Do not use a calculator. - cos⁡16π3\cos \frac { 16 \pi } { 3 }cos316π​

Graph the function. - y=csc⁡(πx5+3π5)y=\csc \left(\frac{\pi x}{5}+\frac{3 \pi}{5}\right)y=csc(5πx​+53π​)

Use the fact that the trigonometric functions are periodic to find the exact value of the expression. Do not use a calculator. - sec⁡19π4\sec \frac { 19 \pi } { 4 }sec419π​

Solve the problem. -For what numbers x,0≤x≤2πx , 0 \leq x \leq 2 \pix,0≤x≤2π , does sin⁡x=−1\sin x = - 1sinx=−1 ?

Functions and Their Graphs

Linear and Quadratic Functions

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Analytic Trigonometry

Applications of Trigonometric Functions

Polar Coordinates; Vectors

Analytic Geometry

Systems of Equations and Inequalities

Sequences; Induction; the Binomial Theorem

Counting and Probability

A Preview of Calculus: the Limit, Derivative, and Integral of a Function

Foundations: a Prelude to Functions

Graphing Utilities

Filters

Name the quadrant in which the angle θ lies. - $\sin \theta > 0 , \quad \cos \theta < 0$

Solve the problem. -An irrigation sprinkler in a field of lettuce sprays water over a distance of 35 feet as it rotates through an angle of $135 ^ { \circ }$ . What area of the field receives water? If necessary, round the answer to two decimal places.

Find the exact value. Do not use a calculator. - $\csc \left( - \frac { \pi } { 2 } \right)$

Solve the problem. -For what numbers $\theta$ is $f ( \theta ) = \cot \theta$ not defined?

Graph the sinusoidal function. - $y=-3 \cos (\pi x)$ $Graph the sinusoidal function. - y=-3 \cos (\pi x)$

Match the function to its graph. - $y = - \tan \left( x - \frac { \pi } { 2 } \right)$

Find the exact value of the expression if θ = 45°. Do not use a calculator. - $f ( \theta ) = \sin \theta \quad$ Find $3 f ( \theta )$ .

Choose the one alternative that best completes the statement or answers the question. Use the even-odd properties to find the exact value of the expression. Do not use a calculator. - $\sin \left( - \frac { \pi } { 4 } \right)$

Use the fact that the trigonometric functions are periodic to find the exact value of the expression. Do not use a calculator. - $\tan 540 ^ { \circ }$

Choose the one alternative that best completes the statement or answers the question. Use transformations to graph the function. - $y=2 \sin (\pi-x)$ $Choose the one alternative that best completes the statement or answers the question. Use transformations to graph the function. - y=2 \sin (\pi-x)$

Draw the angle. - $\frac { 5 \pi } { 3 }$

Draw the angle. - $- \frac { 7 \pi } { 6 }$

Use the fact that the trigonometric functions are periodic to find the exact value of the expression. Do not use a calculator. - $\cos \frac { 16 \pi } { 3 }$

Graph the function. - $y=\csc \left(\frac{\pi x}{5}+\frac{3 \pi}{5}\right)$ $Graph the function. - y=\csc \left(\frac{\pi x}{5}+\frac{3 \pi}{5}\right)$

Use the fact that the trigonometric functions are periodic to find the exact value of the expression. Do not use a calculator. - $\sec \frac { 19 \pi } { 4 }$

Solve the problem. -For what numbers $x , 0 \leq x \leq 2 \pi$ , does $\sin x = - 1$ ?