Deck 11: Trigonometric Functions

Question 1

Evaluate the given expression. sec $\left( - \frac { 17 \pi } { 4 } \right)$

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

Answer 1

$\sqrt { 2 }$

Question 2

Find the indicated integral. $\int x ^ { 5 } \sin \left( 3 + x ^ { 6 } \right) d x$

A) $- \frac { x ^ { 6 } } { 6 } \cos \left( 3 + x ^ { 6 } \right) + C$
B) $\frac { 1 } { 6 } \cos \left( 3 + x ^ { 6 } \right) + C$
C) $- \frac { 1 } { 6 } \cos \left( 3 + x ^ { 6 } \right) + C$
D) $\frac { 1 } { 6 } \sin \left( 3 + x ^ { 6 } \right) + C$

Answer 2

$- \frac { 1 } { 6 } \cos \left( 3 + x ^ { 6 } \right) + C$

Question 3

Find all values of $\theta$ in the specified interval that satisfy the given equation. $\sin \theta = \cos 2 \theta , 0 \leq \theta \leq 2 \pi$

A) $\theta = \frac { \pi } { 6 } , \frac { 5 \pi } { 6 } , \frac { 3 \pi } { 2 }$
B) $\theta = \frac { \pi } { 6 } , \frac { \pi } { 2 } , \frac { 5 \pi } { 6 }$
C) $\theta = \frac { \pi } { 6 } , \frac { \pi } { 2 }$
D) $\theta = \frac { \pi } { 6 } , \frac { 5 \pi } { 6 }$

Answer 3

$\theta = \frac { \pi } { 6 } , \frac { 5 \pi } { 6 } , \frac { 3 \pi } { 2 }$

Question 4

Differentiate the given function. $f ( t ) = \sin 6 t - \cos 4 t$

A) $6 \sin 6 t - 4 \cos 4 t$
B) $6 \cos 6 t + 4 \sin 4 t$
C) $- 6 \cos 6 t + 4 \sin 4 t$
D) $\cos 6 t - \sin 4 t$

Question 5

A manufacturer will supply x units of a certain commodity to the market when the price is $p = S ( x )$ dollars per unit,where $S ( x ) = 23 + 3 x - 2.5 \sin \pi x$ . What is the producers' surplus when x = 5 units?

A)$39.09
B)$113.70
C)$30.18
D)$37.50

Question 6

Solve the given differential equation subject to the specified initial condition. $\frac { d y } { d t } = \frac { \cos t } { 9 y } ; y ( 0 ) = 3$

A) $y ( t ) = 3 - \sqrt { \frac { 2 } { 9 } \sin t }$
B) $y ( t ) = 3 + \sqrt { \frac { 2 } { 9 } \sin t }$
C) $y ( t ) = \sqrt { \frac { 2 } { 9 } \sin t + 9 }$
D) $y ( t ) = \sqrt { 9 - \frac { 2 } { 9 } \sin t }$

Question 7

Find the indicated integral: $\int ( 20 \cos x - 6 \sin x ) d x$

A) $20 \sin x - 6 \cos x + C$
B) $- 20 \sin x + 6 \cos x + C$
C) $20 \sin x + 6 \cos x + C$
D) $- 20 \sin x - 6 \cos x + C$

Question 8

Convert the indicated angle from degrees to radians. $185 ^ { \circ }$

A) $\frac { 37 \pi } { 36 }$
B) $\frac { 37 \pi } { 18 }$
C) $\frac { 37 \pi } { 72 }$
D)None of the above

Question 9

Differentiate the given function. $g ( t ) = 5 \sin ^ { 10 } t - 6 \sin t ^ { 10 }$

A) $50 \sin ^ { 9 } t \cdot \cos t - 60 \sin ^ { 9 } t \cdot \cos t$
B) $- 5 \cos ^ { 10 } t + 6 \cos t ^ { 10 }$
C) $5 \cos ^ { 10 } t - 6 \cos ^ { 10 }$
D) $50 \sin ^ { 9 } t \cdot \cos t - 60 t ^ { 9 } \cos t ^ { 10 }$

Question 10

A projectile is fired from a cannon at ground level at an angle of $24 ^ { \circ }$ with an initial velocity of 54 ft/sec.How far from the cannon does the projectile land? Round your answer to the nearest tenth of a foot.
Use the following formulas
$$\begin{array} { l } x = \left( v _ { 0 } \cos \theta \right) t \\y = - 16 t ^ { 2 } + \left( v _ { 0 } \sin \theta \right) t + s _ { 0 }\end{array}$$

A) 33.9 ft
B) 37.1 ft
C) 67.7 ft
D) 135.4 ft

Question 11

Find the volume of the solid generated by rotating the region R about the x-axis.R is the region bounded by the x-axis, $y = \sec x$ , $x = - \frac { \pi } { 4 }$ ,and $x = \frac { \pi } { 4 }$ .

A) $\frac { 45 \pi } { 2 }$
B) $\frac { 9 \pi } { 2 }$
C) $2 \pi$
D) $9 \pi$

Question 12

Find $\sec \theta$ if $\sin \theta = \frac { 3 } { 5 }$

A) $\frac { 5 } { 4 }$
B) $\frac { 4 } { 5 }$
C) $\frac { 3 } { 4 }$
D) $\frac { 4 } { 3 }$

Question 13

Using a graphing utility,find the value of t for which the smallest value of x(t)occurs for the indicated interval.Estimate to four decimal places,if necessary. $x ( t ) = e ^ { - t } ( 4 \sin 3 t + 3 \cos 2 t ) \text { for } 0 \leq t \leq \pi$

A) $t \approx 0.9605$
B) t=0
C) $t \approx 0.2632$
D)None of the above

Question 14

Specify the number of degrees in the indicated angle. $$\infty$$

A) $150 ^ { \circ }$
B) $210 ^ { \circ }$
C) $240 ^ { \circ }$
D) $- 150 ^ { \circ }$

Question 15

Convert the indicated angle from radians to degrees. $\frac { 23 \pi } { 12 }$

A) $172.5 ^ { \circ }$
B) $690 ^ { \circ }$
C) $345 ^ { \circ }$
D)None of the above

Question 16

Suppose a company determines that the profit P (in hundreds of dollars)obtained from sales during week t (for $0 \leq t \leq 52$ )of a particular year may be modeled by the trigonometric function $P ( t ) = 115 - 40 \cos \left[ \frac { \pi ( t - 17 ) } { 26 } \right]$ What is the largest weekly profit?

A)$15,500
B)$23,000
C)$11,500
D)$7,500

Question 17

Determine the period p,the amplitude b,the phase shift d,and the vertical shift a of the given trigonometric function f(t). $f ( t ) = 2 + 5 \cos \left[ \frac { \pi } { 3 } ( t + 1 ) \right]$

A) $p = 3 , b = - 5 , d = 1 , a = 2$
B) $p = 6 , b = 5 , d = - 1 , a = 2$
C) $p = 3 , b = 5 , d = 1 , a = 2$
D) $p = 6 , b = - 5 , d = 1 , a = 2$

Question 18

Evaluate the given expression. cos $\frac { \pi } { 2 }$

A)Undefined
B)-1
C)0
D)1

Question 19

Evaluate the given expression. tan $\frac { 9 \pi } { 4 }$

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