Question 1

Give the degree measure of .
-ϴ = arccos (0)

Accepted Answer

The arccosine of 0 is 90°, which means the angle whose cosine is 0 is 90 degrees.

Question 2

Solve the equation for exact solutions over the interval [0, 2 $$[ 0,2 \pi )$$
-$\cos x = \sin x$

Accepted Answer

The equation $\cos x = \sin x$ can be solved by using the identity $\sin x = \cos(\frac{\pi}{2} - x)$. Setting $\cos x = \cos(\frac{\pi}{2} - x)$ and solving for $x$ gives the solutions $\frac{\pi}{4}$ and $\frac{5\pi}{4}$ within the interval $[0, 2\pi)$.

Question 3

Provide an appropriate response.
-$$\text { True or false? The statement } \tan \left( \tan ^ { - 1 } \mathrm { x } \right) = \mathrm { x } \text { for all real numbers in the interval } \leq x \leq \pi \text {. }$$

Accepted Answer

The inverse tangent function undoes the tangent function, so $	an(	an^{-1}(x)) = x$ for all real numbers $x$.

Question 4

Solve the equation for solutions in the interval [0, 2 $$[ 0,2 \pi ) \text {. }$$
-$$\sin 4 x = \frac { \sqrt { 3 } } { 2 }$$

Accepted Answer

The equation $\sin 4x = \frac{\sqrt{3}}{2}$ has solutions where the sine function equals $\frac{\sqrt{3}}{2}$, which occurs at angles of $\frac{\pi}{3}$ and $\frac{2\pi}{3}$ in the first revolution. Since the argument of the sine function is $4x$, we divide these angles by 4 to find the solutions in one revolution, and then find all solutions in the interval $[0, 2\pi)$ by considering the periodicity and symmetry of the sine function. This results in eight solutions, as listed in option B.

Question 5

Give the degree measure of .
-$$\theta = \cos ^ { - 1 } \left( \frac { \sqrt { 3 } } { 2 } \right)$$

Accepted Answer

The cosine of 30 degrees is $\sqrt{3}/2$, so the inverse cosine of $\sqrt{3}/2$ is 30 degrees.

Question 6

Solve.
-The output voltage for an $\mathrm { AC }$ generator is approximated by $\mathrm { v } = 156 \cos \left( 120 \pi \mathrm { t } - \frac { \pi } { 3 } \right)$. Find the smallest positive value of $\mathrm { t }$ for which the output is 75 volts. Round values to 4 decimal places.

Accepted Answer

The output voltage is 75 volts when $\cos \left( 120 \pi \mathrm { t } - \frac { \pi } { 3 } \right) = \frac { 75 } { 156 } \approx 0.4808$. Solving for $\mathrm { t }$ gives $\mathrm { t } \approx 0.0056$.

Question 7

Solve the problem.
-The position of a weight on a spring relative to the point of equilibrium is given by y = 4 cos 6t - 2 sin 6t,
Where the arguments are in radians and t is in seconds. Find the smallest value of t for which the
Weight is at the point of equilibrium (y = 0). Give your answer to the nearest hundredth.

Accepted Answer

The answer of Solve the problem.
-The position of a weight...

Question 8

Solve the equation for solutions in the interval [0, 2 $$[ 0,2 \pi )$$
-$2 \sqrt { 3 } \sin 4 x = 3$

Accepted Answer

First, solve for $\sin 4x = \frac{3}{2\sqrt{3}} = \frac{\sqrt{3}}{2}$. The solutions for $\sin \theta = \frac{\sqrt{3}}{2}$ within one cycle [0, $2\pi$] are $\theta = \frac{\pi}{3}$ and $\theta = \frac{2\pi}{3}$. Since we're solving for $4x$, we consider $4x = \frac{\pi}{3} + 2k\pi$ and $4x = \frac{2\pi}{3} + 2k\pi$ for $k = 0, 1, 2, ...$. Solving for $x$ gives the eight solutions listed in option D within the interval [0, $2\pi$).

Question 9

Solve the equation for exact solutions.
-$$- \sin ^ { - 1 } ( 4 x ) = \frac { \pi } { 4 }$$

Accepted Answer

The answer of Solve the equation for exact solutions.
-\[- \sin...

Question 10

Solve the equation for solutions in the interval [0, 2 $$[ 0,2 \pi )$$
-$\cos 2 x = \sqrt { 2 } - \cos 2 x$

Accepted Answer

To solve the equation $\cos 2x = \sqrt{2} - \cos 2x$, first combine like terms to get $2\cos 2x = \sqrt{2}$. Dividing both sides by 2 gives $\cos 2x = \frac{\sqrt{2}}{2}$. The solutions to $\cos \theta = \frac{\sqrt{2}}{2}$ are at $\theta = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4},$ and $\frac{7\pi}{4}$ within one cycle of the cosine function. However, since we have $\cos 2x$, we need to find the values of $x$ that satisfy $2x = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4},$ and $\frac{7\pi}{4}$. Solving for $x$ gives $x = \frac{\pi}{8}, \frac{3\pi}{8}, \frac{5\pi}{8},$ and $\frac{7\pi}{8}$. However, because the equation was $\cos 2x = \sqrt{2} - \cos 2x$, and we solved for $2x$, we must consider the periodicity of the cosine function, which repeats every $2\pi$. Thus, the solutions within the interval $[0, 2\pi)$ are $\frac{\pi}{8}, \frac{9\pi}{8}, \frac{7\pi}{8},$ and $\frac{15\pi}{8}$, which corresponds to choice B.

Question 11

Graph the inverse circular function.
-$y=\tan ^{-1} x$

Accepted Answer

The answer of Graph the inverse circular function.
-$y=\tan ^{-1} x$

...

Question 12

Give the degree measure of .
-ϴ = arcsin (1)

Accepted Answer

The arcsin function, also known as the inverse sine function, of 1 is 90° because sin(90°) = 1.

Question 13

Solve the equation for x, where x is restricted to the given interval.
-$y = \sin x - \pi$, for $x$ in $\left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$

Accepted Answer

The answer of Solve the equation for x, where x...

Question 14

Find the exact value of the real number y.
-$$y = \sin ^ { - 1 } ( 1 )$$

Accepted Answer

The value of $\sin^{-1}(1)$ is the angle whose sine is 1. This angle is $\frac{\pi}{2}$ radians (or 90 degrees), which corresponds to the maximum value of the sine function.

Question 15

Give the degree measure of .
-$$\theta = \arcsin \left( - \frac { 1 } { 2 } \right)$$

Accepted Answer

The arcsin function gives the angle whose sine is the given number. Since the sine of $-30^\circ$ is $-\frac{1}{2}$, $\arcsin\left(-\frac{1}{2}\right) = -30^\circ$.

Question 16

Solve.
-Consider the formula T = 591 - 76 cos 2ϴ, where ϴ is measured in degrees. To the nearest hundredth of a degree, what is the smallest positive value of ϴ for which the value of T will be 577?

Accepted Answer

To solve for ϴ when T = 577, substitute T with 577 in the equation and solve for ϴ:577 = 591 - 76 cos 2ϴSubtract 591 from both sides:-14 = -76 cos 2ϴDivide both sides by -76:cos 2ϴ = 14/76Simplify:cos 2ϴ = 0.18421Take the inverse cosine (in degrees) of both sides:2ϴ = cos⁻¹(0.18421)2ϴ ≈ 79.38°Divide by 2 to solve for ϴ:ϴ ≈ 79.38° / 2ϴ ≈ 39.69°Thus, the smallest positive value of ϴ to the nearest hundredth of a degree is approximately 39.69°.

Question 17

Solve the equation in the interval [0°, 360°). Give solutions to the nearest tenth, if necessary.
-$4 \sin ^ { 2 } \theta = 3$

Accepted Answer

First, solve for $\sin^2 \theta = \frac{3}{4}$, which gives $\sin \theta = \pm \sqrt{\frac{3}{4}} = \pm \frac{\sqrt{3}}{2}$. The solutions for $\sin \theta = \frac{\sqrt{3}}{2}$ are $60^\circ$ and $120^\circ$, and for $\sin \theta = -\frac{\sqrt{3}}{2}$ are $240^\circ$ and $300^\circ$, within the given interval.

Question 18

Solve the equation for x, where x is restricted to the given interval.
-$y = 3 \tan ( 2 x - 1 )$, for $x$ in $\left( \frac { 1 } { 2 } - \frac { \pi } { 4 } , \frac { 1 } { 2 } + \frac { \pi } { 4 } \right)$

Accepted Answer

The equation $y = 3 \tan (2x - 1)$ can be rearranged to solve for $x$ as $x = \frac{1}{2} + \frac{1}{2} \arctan \frac{y}{3}$, which matches option C.

Question 19

Solve the equation for solutions in the interval [0°, 360°). Round to the nearest degree.
-$\sqrt { 3 } \sec 2 \theta = 2$

Accepted Answer

The answer of Solve the equation for solutions in the...

Question 20

Solve the equation for x, where x is restricted to the given interval.
-$$y = 7 \sin x \text {, for } x \text { in } \left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$$

Accepted Answer

To solve for $x$, we isolate $x$ on one side of the equation: $x = \arcsin \frac { y } { 7 }$. This is the inverse operation of sine, applied to both sides of the equation, correctly isolating $x$.

Quiz 6: Inverse Circular Functions and Trigonometric Equations

Give the degree measure of . -ϴ = arccos (0)

Solve the equation for exact solutions over the interval [0, 2 $[ 0,2 \pi )$ - $\cos x = \sin x$

Provide an appropriate response. - $\text { True or false? The statement } \tan \left( \tan ^ { - 1 } \mathrm { x } \right) = \mathrm { x } \text { for all real numbers in the interval } \leq x \leq \pi \text {. }$

Solve the equation for solutions in the interval [0, 2 $[ 0,2 \pi ) \text {. }$ - $\sin 4 x = \frac { \sqrt { 3 } } { 2 }$

Give the degree measure of . - $\theta = \cos ^ { - 1 } \left( \frac { \sqrt { 3 } } { 2 } \right)$

Solve. -The output voltage for an $\mathrm { AC }$ generator is approximated by $\mathrm { v } = 156 \cos \left( 120 \pi \mathrm { t } - \frac { \pi } { 3 } \right)$ . Find the smallest positive value of $\mathrm { t }$ for which the output is 75 volts. Round values to 4 decimal places.

Solve the equation for solutions in the interval [0, 2 $[ 0,2 \pi )$ - $2 \sqrt { 3 } \sin 4 x = 3$

Solve the equation for exact solutions. - $- \sin ^ { - 1 } ( 4 x ) = \frac { \pi } { 4 }$

Solve the equation for solutions in the interval [0, 2 $[ 0,2 \pi )$ - $\cos 2 x = \sqrt { 2 } - \cos 2 x$

Graph the inverse circular function. - $y=\tan ^{-1} x$ $Graph the inverse circular function. - y=\tan ^{-1} x A) B) C) D)$

Give the degree measure of . -ϴ = arcsin (1)

Solve the equation for x, where x is restricted to the given interval. - $y = \sin x - \pi$ , for $x$ in $\left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$

Find the exact value of the real number y. - $y = \sin ^ { - 1 } ( 1 )$

Give the degree measure of . - $\theta = \arcsin \left( - \frac { 1 } { 2 } \right)$

Solve. -Consider the formula T = 591 - 76 cos 2ϴ, where ϴ is measured in degrees. To the nearest hundredth of a degree, what is the smallest positive value of ϴ for which the value of T will be 577?

Solve the equation in the interval [0°, 360°) . Give solutions to the nearest tenth, if necessary. - $4 \sin ^ { 2 } \theta = 3$

Solve the equation for x, where x is restricted to the given interval. - $y = 3 \tan ( 2 x - 1 )$ , for $x$ in $\left( \frac { 1 } { 2 } - \frac { \pi } { 4 } , \frac { 1 } { 2 } + \frac { \pi } { 4 } \right)$

Solve the equation for solutions in the interval [0°, 360°) . Round to the nearest degree. - $\sqrt { 3 } \sec 2 \theta = 2$

Solve the equation for x, where x is restricted to the given interval. - $y = 7 \sin x \text {, for } x \text { in } \left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$

Quiz 6: Inverse Circular Functions and Trigonometric Equations

Give the degree measure of . -ϴ = arccos (0)

Solve the equation for exact solutions over the interval [0, 2 [0,2π) [ 0,2 \pi ) [0,2π) - cos⁡x=sin⁡x\cos x = \sin xcosx=sinx

Solve the equation for solutions in the interval [0, 2 [0,2π) . [ 0,2 \pi ) \text {. }[0,2π) . - sin⁡4x=32\sin 4 x = \frac { \sqrt { 3 } } { 2 }sin4x=23​​

Give the degree measure of . - θ=cos⁡−1(32) \theta = \cos ^ { - 1 } \left( \frac { \sqrt { 3 } } { 2 } \right) θ=cos−1(23​​)

Solve the equation for solutions in the interval [0, 2 [0,2π) [ 0,2 \pi ) [0,2π) - 23sin⁡4x=32 \sqrt { 3 } \sin 4 x = 323​sin4x=3

Solve the equation for exact solutions. - −sin⁡−1(4x) =π4- \sin ^ { - 1 } ( 4 x ) = \frac { \pi } { 4 }−sin−1(4x) =4π​

Solve the equation for solutions in the interval [0, 2 [0,2π) [ 0,2 \pi ) [0,2π) - cos⁡2x=2−cos⁡2x\cos 2 x = \sqrt { 2 } - \cos 2 xcos2x=2​−cos2x

Graph the inverse circular function. - y=tan⁡−1xy=\tan ^{-1} xy=tan−1x

Give the degree measure of . -ϴ = arcsin (1)

Solve the equation for x, where x is restricted to the given interval. - y=sin⁡x−πy = \sin x - \piy=sinx−π , for xxx in [−π2,π2]\left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right][−2π​,2π​]

Find the exact value of the real number y. - y=sin⁡−1(1) y = \sin ^ { - 1 } ( 1 ) y=sin−1(1)

Give the degree measure of . - θ=arcsin⁡(−12) \theta = \arcsin \left( - \frac { 1 } { 2 } \right) θ=arcsin(−21​)