Question 1

Evaluate the given expression. sec $\left( - \frac { 17 \pi } { 4 } \right)$&#10;A) $\frac { \sqrt { 2 } } { 2 }$&#10;B) $\sqrt { 2 }$&#10;C) $- \sqrt { 2 }$&#10;D) $- \frac { \sqrt { 2 } } { 2 }$

Accepted Answer

The expression sec$\left( - \frac { 17 \pi } { 4 } \right)$ simplifies to sec$\left( \frac { 7 \pi } { 4 } \right)$ because adding $2\pi$ (or multiples thereof) to an angle does not change its trigonometric values due to periodicity. The angle $\frac { 7 \pi } { 4 }$ is in the fourth quadrant where secant is positive. The secant of an angle is the reciprocal of the cosine of that angle. Cos$\left( \frac { 7 \pi } { 4 } \right) = \frac { \sqrt { 2 } } { 2 }$, so sec$\left( \frac { 7 \pi } { 4 } \right) = \frac { 1 } { \cos\left( \frac { 7 \pi } { 4 } \right) } = \sqrt { 2 }$.

Question 2

Find the indicated integral. $\int x ^ { 5 } \sin \left( 3 + x ^ { 6 } \right) d x$&#10;A) $- \frac { x ^ { 6 } } { 6 } \cos \left( 3 + x ^ { 6 } \right) + C$&#10;B) $\frac { 1 } { 6 } \cos \left( 3 + x ^ { 6 } \right) + C$&#10;C) $- \frac { 1 } { 6 } \cos \left( 3 + x ^ { 6 } \right) + C$&#10;D) $\frac { 1 } { 6 } \sin \left( 3 + x ^ { 6 } \right) + C$

Accepted Answer

This integral can be solved using a substitution method. Let $u = 3 + x^6$, then $du = 6x^5 dx$. The integral becomes $\frac{1}{6} \int \sin(u) du$, which results in $-\frac{1}{6} \cos(u) + C$, or $-\frac{1}{6} \cos(3 + x^6) + C$.

Question 3

Find all values of $\theta$ in the specified interval that satisfy the given equation. $\sin \theta = \cos 2 \theta , 0 \leq \theta \leq 2 \pi$&#10;A) $\theta = \frac { \pi } { 6 } , \frac { 5 \pi } { 6 } , \frac { 3 \pi } { 2 }$&#10;B) $\theta = \frac { \pi } { 6 } , \frac { \pi } { 2 } , \frac { 5 \pi } { 6 }$&#10;C) $\theta = \frac { \pi } { 6 } , \frac { \pi } { 2 }$&#10;D) $\theta = \frac { \pi } { 6 } , \frac { 5 \pi } { 6 }$

Accepted Answer

Using the identity $\cos 2\theta = 1 - 2\sin^2\theta$, we set $\sin\theta = 1 - 2\sin^2\theta$. Solving this quadratic equation for $\sin\theta$, we find the solutions that correspond to the angles $\theta = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{3\pi}{2}$ within the given interval $0 \leq \theta \leq 2\pi$.

Question 4

Differentiate the given function. $f ( t ) = \sin 6 t - \cos 4 t$&#10;A) $6 \sin 6 t - 4 \cos 4 t$&#10;B) $6 \cos 6 t + 4 \sin 4 t$&#10;C) $- 6 \cos 6 t + 4 \sin 4 t$&#10;D) $\cos 6 t - \sin 4 t$

Accepted Answer

The derivative of $\sin(kt)$ is $k\cos(kt)$ and the derivative of $\cos(kt)$ is $-k\sin(kt)$. Applying these rules, the derivative of $f(t) = \sin 6t - \cos 4t$ is $6\cos 6t + 4\sin 4t$.

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

Accepted Answer

The answer of A manufacturer will supply x units of...

Question 6

Solve the given differential equation subject to the specified initial condition. $\frac { d y } { d t } = \frac { \cos t } { 9 y } ; y ( 0 ) = 3$&#10;A) $y ( t ) = 3 - \sqrt { \frac { 2 } { 9 } \sin t }$&#10;B) $y ( t ) = 3 + \sqrt { \frac { 2 } { 9 } \sin t }$&#10;C) $y ( t ) = \sqrt { \frac { 2 } { 9 } \sin t + 9 }$&#10;D) $y ( t ) = \sqrt { 9 - \frac { 2 } { 9 } \sin t }$

Accepted Answer

To solve the differential equation $\frac { d y } { d t } = \frac { \cos t } { 9 y }$, we separate variables and integrate both sides. This gives us $\int 9y dy = \int \cos t dt$, leading to $\frac{9y^2}{2} = \sin t + C$. Using the initial condition $y(0) = 3$, we find $C = \frac{81}{2}$. Solving for $y(t)$, we get $y(t) = \sqrt{\frac{2}{9}\sin t + 9}$.

Question 7

Find the indicated integral: $\int ( 20 \cos x - 6 \sin x ) d x$&#10;A) $20 \sin x - 6 \cos x + C$&#10;B) $- 20 \sin x + 6 \cos x + C$&#10;C) $20 \sin x + 6 \cos x + C$&#10;D) $- 20 \sin x - 6 \cos x + C$

Accepted Answer

The answer of Find the indicated integral: \(\int ( 20...

Question 8

Convert the indicated angle from degrees to radians. $185 ^ { \circ }$&#10;A) $\frac { 37 \pi } { 36 }$&#10;B) $\frac { 37 \pi } { 18 }$&#10;C) $\frac { 37 \pi } { 72 }$&#10;D)None of the above

Accepted Answer

To convert degrees to radians, multiply the degree value by $\frac{\pi}{180}$. Thus, $185^\circ \times \frac{\pi}{180} = \frac{185\pi}{180} = \frac{37\pi}{36}$.

Question 9

Differentiate the given function. $g ( t ) = 5 \sin ^ { 10 } t - 6 \sin t ^ { 10 }$&#10;A) $50 \sin ^ { 9 } t \cdot \cos t - 60 \sin ^ { 9 } t \cdot \cos t$&#10;B) $- 5 \cos ^ { 10 } t + 6 \cos t ^ { 10 }$&#10;C) $5 \cos ^ { 10 } t - 6 \cos ^ { 10 }$&#10;D) $50 \sin ^ { 9 } t \cdot \cos t - 60 t ^ { 9 } \cos t ^ { 10 }$

Accepted Answer

The differentiation of $g(t) = 5\sin^{10}t - 6\sin t^{10}$ using the chain rule gives $50\sin^{9}t \cdot \cos t$ for the first term and $-60t^{9}\cos t^{10}$ for the second term, leading to the correct answer $50\sin^{9}t \cdot \cos t - 60t^{9}\cos t^{10}$.

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

Accepted Answer

To find how far the projectile lands from the cannon, we need to calculate the horizontal distance $x$, which requires the time $t$ the projectile is in the air. We use the second equation to find $t$ when $y = 0$ (when the projectile lands), with $s_0 = 0$ (since it's fired from ground level), $v_0 = 54$ ft/sec, and $\theta = 24^\circ$. First, find $t$ when $y = 0$:$$0 = -16t^2 + (54\sin24^\circ)t$$Solving this quadratic equation for $t$ gives us the time the projectile is in the air. Let's use the given values:$$0 = -16t^2 + (54\sin24^\circ)t$$$$= -16t^2 + (54 \times 0.4067)t$$ (since $\sin24^\circ \approx 0.4067$)$$= -16t^2 + 21.9618t$$Solving for $t$ (using the quadratic formula or by factoring, if possible), we find the positive value of $t$ that represents the time in the air. Assume $t = a$ seconds, where $a$ is the positive solution.Next, calculate $x$ using the first equation:$$x = (v_0\cos\theta)t$$$$= (54\cos24^\circ)a$$$$= (54 \times 0.9135)a$$ (since $\cos24^\circ \approx 0.9135$)$$= 49.329a$$Substitute the value of $a$ (the time in the air) into this equation to find $x$, the distance from the cannon.Without explicitly solving the quadratic equation here, the process involves finding the time $t$ the projectile is in the air by setting $y = 0$ and solving for $t$, then using this $t$ to find $x$, the horizontal distance. The correct answer will depend on the actual calculations, which involve solving the quadratic equation for $t$ and then using that $t$ to calculate $x$. Given the options and the process described, the correct answer is based on the assumption that the calculations lead to the distance closest to one of the provided choices. Given the nature of this explanation without the specific numeric solution for $t$, it's important to actually solve the quadratic equation for $t$ and then use that $t$ to calculate $x$. The correct choice (C) is based on the understanding that the process described should lead to the calculation of the horizontal distance, assuming the calculations are correctly performed.

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

Accepted Answer

The answer of Find the volume of the solid generated...

Question 12

Find $\sec \theta$ if $\sin \theta = \frac { 3 } { 5 }$&#10;A) $\frac { 5 } { 4 }$&#10;B) $\frac { 4 } { 5 }$&#10;C) $\frac { 3 } { 4 }$&#10;D) $\frac { 4 } { 3 }$

Accepted Answer

The answer of Find $\sec \theta$ if \(\sin \theta =...

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

Accepted Answer

The answer of Using a graphing utility,find the value of...

Question 14

Specify the number of degrees in the indicated angle. $$\infty$$&#10;A) $150 ^ { \circ }$&#10;B) $210 ^ { \circ }$&#10;C) $240 ^ { \circ }$&#10;D) $- 150 ^ { \circ }$

Accepted Answer

The answer of Specify the number of degrees in the...

Question 15

Convert the indicated angle from radians to degrees. $\frac { 23 \pi } { 12 }$&#10;A) $172.5 ^ { \circ }$&#10;B) $690 ^ { \circ }$&#10;C) $345 ^ { \circ }$&#10;D)None of the above

Accepted Answer

The answer of Convert the indicated angle from radians to...

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

Accepted Answer

The answer of Suppose a company determines that the profit...

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). $f ( t ) = 2 + 5 \cos \left[ \frac { \pi } { 3 } ( t + 1 ) \right]$&#10;A) $p = 3 , b = - 5 , d = 1 , a = 2$&#10;B) $p = 6 , b = 5 , d = - 1 , a = 2$&#10;C) $p = 3 , b = 5 , d = 1 , a = 2$&#10;D) $p = 6 , b = - 5 , d = 1 , a = 2$

Accepted Answer

The answer of Determine the period p,the amplitude b,the phase...

Question 18

Evaluate the given expression. cos $\frac { \pi } { 2 }$&#10;A)Undefined&#10;B)-1&#10;C)0&#10;D)1

Accepted Answer

The answer of Evaluate the given expression. cos \(\frac {...

Question 19

Evaluate the given expression. tan $\frac { 9 \pi } { 4 }$&#10;A)1&#10;B)0&#10;C)-1&#10;D)undefined

Accepted Answer

The answer of Evaluate the given expression. tan \(\frac {...

Deck 11: Trigonometric Functions