1.
Item 6. An angle of \(225^\circ\) expressed in radians is:
Convert degrees to radians using \(180^\circ=\pi\) radians. \(225^\circ\cdot\frac{\pi}{180^\circ}=\frac{225\pi}{180}\) Simplify the fraction. \(\frac{225\pi}{180}=\frac{5\pi}{4}\) Answer: D
2.
Item 10. What is the \(x\)-coordinate of the point on the unit circle at an angle of \(\frac{\pi}{3}\)?
On the unit circle, the \(x\)-coordinate is \(\cos\theta\). Here \(\theta=\frac{\pi}{3}\). \(x=\cos\left(\frac{\pi}{3}\right)\) \(\cos\left(\frac{\pi}{3}\right)=\frac{1}{2}\) Answer: E
3.
Item 1. A right-angle triangle has a hypotenuse of \(25\) inches. If one angle has a measure of \(49^\circ\), what is the measure of the side opposite that angle? Give your answer to one decimal place.
Use the sine ratio because the opposite side and hypotenuse are involved. \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\) \(\sin 49^\circ=\frac{x}{25}\) \(x=25\sin 49^\circ\) \(x\approx25(0.7547)\) \(x\approx18.9\text{ in}\) Answer: B
4.
Item 8. Which pair of angles are coterminal with \(\frac{5\pi}{3}\) radians?
Coterminal angles differ by integer multiples of \(2\pi\). Start with \(\frac{5\pi}{3}\). \(\frac{5\pi}{3}+2\pi=\frac{5\pi}{3}+\frac{6\pi}{3}=\frac{11\pi}{3}\) \(\frac{5\pi}{3}-2\pi=\frac{5\pi}{3}-\frac{6\pi}{3}=-\frac{\pi}{3}\) Therefore, the coterminal pair is \(-\frac{\pi}{3}\) and \(\frac{11\pi}{3}\). Answer: C
5.
Item 2. If the adjacent side to an acute angle of a right triangle measures \(6.8\) feet and the hypotenuse measures \(9.1\) feet, determine the measure of the acute angle to two decimal places.
Use the cosine ratio because the adjacent side and hypotenuse are given. \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\) \(\cos\theta=\frac{6.8}{9.1}\) \(\theta=\cos^{-1}\left(\frac{6.8}{9.1}\right)\) \(\theta\approx41.65^\circ\) Answer: E
6.
Item 7. The radian measure of an angle is \(\frac{3\pi}{4}\). The arc that subtends the angle has a length of \(6\pi\). Determine the radius of the circle.
Use the arc length formula \(a=r\theta\). Here \(a=6\pi\) and \(\theta=\frac{3\pi}{4}\). \(6\pi=r\left(\frac{3\pi}{4}\right)\) \(r=\frac{6\pi}{\frac{3\pi}{4}}\) \(r=6\pi\cdot\frac{4}{3\pi}=8\) Answer: A
7.
Item 16. Of the following equations, which have undergone a horizontal contraction from their original equations \(y=\sin x\) and \(y=\cos x\)? I. \(y=\cos 3x\) II. \(y=\sin\left(\frac{3}{2}x\right)\) III. \(y=\cos\left(\frac{1}{4}x\right)\) IV. \(y=\sin\left(\frac{2}{3}x\right)\) V. \(y=\cos\left(\frac{4}{3}x\right)\)
For \(y=\sin bx\) or \(y=\cos bx\), a horizontal contraction occurs when \(|b|>1\). I. \(b=3>1\), so this is a contraction. II. \(b=\frac{3}{2}>1\), so this is a contraction. III. \(b=\frac{1}{4}<1\), so this is an expansion. IV. \(b=\frac{2}{3}1\), so this is a contraction. Therefore, I, II, and V have undergone horizontal contraction. Answer: B
8.
Item 9. Determine the exact value of \(\sin\left(\frac{\pi}{4}\right)+\tan\left(\frac{5\pi}{4}\right)\).
\(\sin\left(\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}\). The angle \(\frac{5\pi}{4}\) is in Quadrant III with reference angle \(\frac{\pi}{4}\). Tangent is positive in Quadrant III, so \(\tan\left(\frac{5\pi}{4}\right)=1\). \(\sin\left(\frac{\pi}{4}\right)+\tan\left(\frac{5\pi}{4}\right)=\frac{\sqrt{2}}{2}+1\) \(\frac{\sqrt{2}}{2}+1=\frac{\sqrt{2}}{2}+\frac{2}{2}=\frac{2+\sqrt{2}}{2}\) Answer: B
9.
Item 12. Which of the following shows one period of the graph of \(f(x)=2\sin(3x)\)? [Graphs will be inserted later.]
The function \(f(x)=2\sin(3x)\) has amplitude \(2\). The period is \(\frac{2\pi}{b}\). Here \(b=3\). \(\text{Period}=\frac{2\pi}{3}\). The correct graph must show one full sine cycle with amplitude \(2\) over a length of \(\frac{2\pi}{3}\). Graph D matches these features. Answer: D
10.
Item 3. Solve this right triangle. Give the measures to the nearest tenth. [Diagram will be inserted later.]
Use the Pythagorean theorem to find \(JL\). \(c^2=a^2+b^2\) \(c^2=23^2+9^2=529+81=610\) \(c=\sqrt{610}\approx24.7\text{ cm}\) Next, use tangent to find \(\angle J\). \(\tan J=\frac{9}{23}\) \(J=\tan^{-1}\left(\frac{9}{23}\right)\approx21.4^\circ\) The two acute angles of a right triangle add to \(90^\circ\). \(\angle L=90^\circ-21.4^\circ=68.6^\circ\) Answer: B
11.
Item 4. Determine the reference angle for \(215^\circ\) in standard position.
The angle \(215^\circ\) is in Quadrant III. The reference angle is the positive acute angle between the terminal arm and the \(x\)-axis. In Quadrant III, subtract \(180^\circ\). \(\text{Reference Angle}=215^\circ-180^\circ=35^\circ\) Answer: A
12.
Item 14. Determine the horizontal translation of the function \(y=5\sin\left(\frac{1}{2}x-6\right)\).
Factor the coefficient of \(x\) inside the sine function. \(y=5\sin\left(\frac{1}{2}x-6\right)\) \(y=5\sin\left(\frac{1}{2}(x-12)\right)\) The graph is translated \(12\) units to the right. Answer: B
13.
Item 18. Which of the following has a period of \(5\)?
For \(y=\sin(bx)\), the period is \(\frac{2\pi}{|b|}\). To have period \(5\), use \(b=\frac{2\pi}{5}\). Option D is \(y=\sin\left(\frac{2\pi x}{5}\right)\). Its period is \(\frac{2\pi}{\frac{2\pi}{5}}=5\). Answer: D
14.
Item 5. Which of the following is a reasonable measure for angle \(\theta\) shown below in standard position? [Diagram will be inserted later.]
The angle starts on the positive \(x\)-axis, which is the initial arm of an angle in standard position. The terminal arm rotates clockwise, so the angle is negative. The angle shown is between \(-270^\circ\) and \(-360^\circ\). Therefore, the most reasonable estimate is \(-315^\circ\). Answer: C
15.
Item 13. Visualize the graph of \(\cos\theta\) for \(-2\pi\leq\theta\leq2\pi\). What is the range of this graph?
The cosine function has a maximum value of \(1\) and a minimum value of \(-1\). Therefore, the range is \(-1\leq y\leq1\). Answer: C
16.
Item 15. The range of the trigonometric function \(y=a\cos x+b\) is \(-2\leq y\leq8\). Determine the value of \(b\).
The value \(b\) is the equilibrium line or vertical displacement. \(b=\frac{\text{max}+\text{min}}{2}\) \(b=\frac{8+(-2)}{2}\) \(b=\frac{6}{2}=3\) Answer: C
17.
Item 11. Visualize the graph of \(\sin\theta\) for \(-2\pi\leq\theta\leq2\pi\). For what values of \(\theta\) is the graph at the maximum value?
The maximum value of \(\sin\theta\) is \(1\). On the interval \(-2\pi\leq\theta\leq2\pi\), \(\sin\theta=1\) at: \(\theta=-\frac{3\pi}{2}\) and \(\theta=\frac{\pi}{2}\). Answer: A
18.
Item 20. Where are the asymptotes of \(y=\sin\left(x-\frac{\pi}{5}\right)\)?
A sine function does not have vertical asymptotes. The function \(y=\sin\left(x-\frac{\pi}{5}\right)\) is defined for all real values of \(x\). Therefore, there are no asymptotes. Answer: B
19.
Item 17. Given \(y=\cos 4x\), what is the period of this function?
For \(y=\cos nx\), the period is \(\frac{2\pi}{|n|}\). Here \(n=4\). \(\text{Period}=\frac{2\pi}{4}=\frac{\pi}{2}\) Answer: D
20.
Item 19. At a seaport, the depth of the water, \(d\), in metres, at time \(t\) hours, during a certain day is given by: \(d=5.2\sin\left(\frac{2\pi(t-6)}{12.4}\right)+7.8\). On that day, determine the depth of the water at 4:30 PM.
Convert \(4{:}30\text{ PM}\) to 24-hour decimal time. \(4\text{ PM}=16\) hours, and \(30\) minutes is \(0.5\) hours. So \(t=16.5\). Substitute into the equation: \(d=5.2\sin\left(\frac{2\pi(16.5-6)}{12.4}\right)+7.8\) \(d\approx5.2\sin(5.3204)+7.8\) \(d\approx5.2(-0.82076)+7.8\) \(d\approx3.53\text{ m}\) Answer: D
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