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Item 1. Formula Sheet-PC Math 12. Below is the formula sheet for this course, as found in the course intro. You will be provided this within all quizzes and tests for your use. You can click back to this location to make use of the information as needed throughout the test.
This item is a formula sheet download/reference item. No calculation is required for this item. Use the formula sheet as needed throughout the quiz. Answer: N/N
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Item 4. Calculate the arc length that subtends an angle of \(2.32\) radians at the center of a circle with a radius of \(15\text{ m}\).
Use the arc length formula \(a=r\theta\), where \(\theta\) is measured in radians. Here \(r=15\) and \(\theta=2.32\). \(a=15(2.32)\) \(a=34.8\text{ m}\) Therefore the arc length is \(34.8\text{ m}\). Answer: C
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Item 5. What is the \(y\)-coordinate of the point on the unit circle at an angle of \(\frac{5\pi}{4}\)?
On the unit circle, the \(y\)-coordinate is \(\sin\theta\). Here \(\theta=\frac{5\pi}{4}\). The angle \(\frac{5\pi}{4}\) is in Quadrant III. The reference angle is \(\frac{\pi}{4}\), and sine is negative in Quadrant III. \(\sin\left(\frac{5\pi}{4}\right)=-\frac{\sqrt{2}}{2}\). Therefore the \(y\)-coordinate is \(-\frac{\sqrt{2}}{2}\). Answer: C
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Item 6. 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 \(\theta=\frac{\pi}{2}\). Therefore the graph is at its maximum value at \(-\frac{3\pi}{2}\) and \(\frac{\pi}{2}\). Answer: E
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Item 2. A right-angle triangle has a hypotenuse of \(37\text{ cm}\). If one angle has a measure of \(21^\circ\), 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. \(\sin A=\frac{\text{opposite}}{\text{hypotenuse}}\) \(\sin 21^\circ=\frac{\text{opposite}}{37}\) \(\text{opposite}=37\sin 21^\circ\) \(\text{opposite}\approx 13.3\text{ cm}\) The calculated value matches the \(13.3\text{ cm}\) option. The provided answer key marks this item as \(E\), so keep the original answer as \(E\). Answer: E
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Item 3. Determine the reference angle for \(150^\circ\) in standard position.
The angle \(150^\circ\) is in Quadrant II. The reference angle is the positive acute angle between the terminal arm and the \(x\)-axis. In Quadrant II, \(\text{Reference Angle}=180^\circ-\theta\). \(\text{Reference Angle}=180^\circ-150^\circ=30^\circ\). Therefore the reference angle is \(30^\circ\). Answer: D
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Item 7. Visualize the graph of \(\cos\theta\) for \(-2\pi\leq\theta\leq2\pi\). For what values of \(\theta\) is the graph at the maximum value?
The maximum value of \(\cos\theta\) is \(1\). On the interval \(-2\pi\leq\theta\leq2\pi\), \(\cos\theta=1\) at: \(\theta=-2\pi, 0, 2\pi\). Therefore the graph is at its maximum value at \(-2\pi, 0,\) and \(2\pi\). Answer: E
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