1.
Item 2. Determine the reference angle for \(45^\circ\) in standard position.
The angle \(45^\circ\) is in Quadrant I. The reference angle is the positive acute angle between the terminal arm and the \(x\)-axis. In Quadrant I, the reference angle equals the angle in standard position. Therefore the reference angle is \(45^\circ\). Answer: A
2.
Item 4. What is the \(x\)-coordinate of the point on the unit circle at an angle of \(\frac{\pi}{6}\)?
On the unit circle, the \(x\)-coordinate is \(\cos\theta\). Here \(\theta=\frac{\pi}{6}\). \(x=\cos\left(\frac{\pi}{6}\right)\) Using the special triangle values, \(\cos\left(\frac{\pi}{6}\right)=\frac{\sqrt{3}}{2}\). Therefore the \(x\)-coordinate is \(\frac{\sqrt{3}}{2}\). Answer: H
3.
Item 1. Find \(\theta\) if \(x=5\) and \(z=4\). [Diagram will be inserted later.]
Use the cosine ratio because the adjacent side and hypotenuse are given. \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\) \(\cos\theta=\frac{z}{x}\) \(\cos\theta=\frac{4}{5}\) \(\theta=\cos^{-1}\left(\frac{4}{5}\right)\) \(\theta\approx 36.9^\circ\) Rounded to the nearest degree, \(\theta\approx 37^\circ\). Answer: C
4.
Item 3. Which of the following is a reasonable measure for angle \(\theta\) shown below in standard position?
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 diagram shows the terminal arm after a clockwise rotation between \(-270^\circ\) and \(-360^\circ\). The reasonable estimate is \(-315^\circ\). Answer: B
5.
Item 5. Consider the graph of \(y=\sin\theta\) where \(-2\pi\leq \theta\leq 2\pi\). What is the range of this graph?
The sine function has a maximum value of \(1\) and a minimum value of \(-1\). This means the output values of \(y=\sin\theta\) satisfy \(-1\leq y\leq 1\). The interval \(-2\pi\leq \theta\leq 2\pi\) includes full cycles of sine, so the graph reaches both \(-1\) and \(1\). Therefore the range is \(-1\) to \(1\). Answer: B
6.
Item 6. Consider the graph of \(y=\cos\theta\) where \(-2\pi\leq \theta\leq 2\pi\). What is the range of this graph?
The cosine function has a maximum value of \(1\) and a minimum value of \(-1\). This means the output values of \(y=\cos\theta\) satisfy \(-1\leq y\leq 1\). The interval \(-2\pi\leq \theta\leq 2\pi\) includes full cycles of cosine, so the graph reaches both \(-1\) and \(1\). Therefore the range is \(-1\) to \(1\). Answer: B
1 out of 1
