Precalculus Quiz: Extending Trigonometric Functions With Unit Circle

On the unit circle in the coordinate plane, an angle of $\theta=\frac{3\pi}{2}$ radians is drawn in standard position. The terminal point is $P(x,y)$ where $\cos(\theta)=x$ and $\sin(\theta)=y$, and $\tan(\theta)=\frac{y}{x}$ when defined. For the angle described, what is the value of $\tan(\theta)$?

The unit circle enables us to define trigonometric functions for all real numbers. Consider the function $f(x) = \sin(x) + \cos\left(x + \frac{3\pi}{2}\right)$. What is the value of $f\left(\frac{5\pi}{6}\right)$?

On the unit circle in the coordinate plane, an angle of $\theta=\frac{3\pi}{2}$ radians is drawn in standard position. The terminal point is $P(x,y)$ where $\cos(\theta)=x$ and $\sin(\theta)=y$, and $\tan(\theta)=\frac{y}{x}$ when defined. For the angle described, what is the value of $\tan(\theta)$?​

An angle of $\theta=\frac{9\pi}{4}$ radians is drawn in standard position on the unit circle in the coordinate plane. (This is one full rotation of $2\pi$ plus an additional $\frac{\pi}{4}$.) For the angle described, which coordinates represent the terminal point of this angle on the unit circle?

On the unit circle in the coordinate plane, point $P$ is located at coordinates $\left(-\frac{1}{2},\frac{\sqrt{3}}{2}\right)$. An angle $\theta$ in standard position has its terminal side passing through $P$. Using the given information, what is the value of $\sin(\theta)$?

On the unit circle, point $P$ is located at coordinates $\left(-\frac{1}{2},\frac{\sqrt{3}}{2}\right)$. An angle $\theta$ in standard position has its terminal side passing through $P$, so that $\cos(\theta)=x$ and $\sin(\theta)=y$. Based on the unit circle, what is the value of $\sin(\theta)$?

An angle of $\theta=-\frac{\pi}{3}$ radians is drawn in standard position on the unit circle in the coordinate plane (measured clockwise from the positive $x$-axis). For the angle described, what is the value of $\sin(\theta)$?​

An angle of $\frac{9\pi}{4}$ radians is drawn in standard position on the unit circle in the coordinate plane. Since angles that differ by $2\pi$ are co-terminal, $\frac{9\pi}{4}$ has the same terminal point as an angle in Quadrant I. For the angle described, what is the value of $\cos\left(\frac{9\pi}{4}\right)$?​​​

On the unit circle in the coordinate plane, an angle of $\theta=\frac{5\pi}{6}$ radians is drawn in standard position (measured counterclockwise from the positive $x$-axis). Using the fact that a point $(x,y)$ on the unit circle at angle $\theta$ satisfies $\cos(\theta)=x$ and $\sin(\theta)=y$, what is the value of $\cos(\theta)$?

An angle $\theta$ in standard position terminates in Quadrant III on the unit circle and has reference angle $\frac{\pi}{4}$. Based on the unit circle, what is the value of $\tan(\theta)$?

An angle of $\theta=\frac{9\pi}{4}$ is drawn in standard position on the unit circle (one full rotation plus an additional $\frac{\pi}{4}$). For the angle described, what is the value of $\sin(\theta)$?

On the unit circle in the coordinate plane, an angle $\theta=\frac{5\pi}{6}$ is drawn in standard position (measured counterclockwise from the positive $x$-axis). For the angle described, what is the value of $\cos(\theta)$?

An angle $\theta$ is drawn in standard position on the unit circle in the coordinate plane with measure $\theta=\frac{7\pi}{6}$. The terminal point on the unit circle is $P(x,y)$, where $\cos(\theta)=x$ and $\sin(\theta)=y$. For the angle described, what is the value of $\sin(\theta)$?​​​

An angle of $\theta=-\frac{\pi}{3}$ radians is drawn in standard position on the unit circle in the coordinate plane (measured clockwise from the positive $x$-axis). For the angle described, what is the value of $\sin(\theta)$?

An angle $\theta$ in standard position terminates in Quadrant III on the unit circle and has reference angle $\frac{\pi}{4}$. Based on the unit circle, what is the value of $\tan(\theta)$? (Recall $\tan(\theta)=\frac{y}{x}$.)

Using the unit circle, consider angles $\alpha$ and $\beta$ where $\alpha = \frac{11\pi}{6}$ and $\beta = -\frac{\pi}{6}$. A student claims that $\sin(\alpha) \neq \sin(\beta)$ because the angles have different signs. Based on the unit circle interpretation, which statement best evaluates this claim?

The unit circle allows trigonometric functions to be defined for any real number input. If point $P$ on the unit circle corresponds to angle $t$ where $\cos(t) = \frac{2}{3}$, what is the value of $\cos(t + \pi) + \cos(t + 3\pi)$?

On the unit circle, point $P$ is located at coordinates $\left(-\frac{1}{2},\frac{\sqrt{3}}{2}\right)$. An angle $\theta$ in standard position has its terminal side passing through $P$. Based on the unit circle definition $\sin(\theta)=y$, what is the value of $\sin(\theta)$?

Two angles, $\alpha=\frac{\pi}{3}$ and $\beta=\frac{5\pi}{3}$, are drawn in standard position on the unit circle in the coordinate plane. Using the unit circle definition $\sin(\theta)=y$ and quadrant sign patterns, what is the sign of $\sin(\beta)$?​

An angle $\theta$ is drawn in standard position on the unit circle in the coordinate plane with $\theta=\frac{11\pi}{6}$. The terminal point on the unit circle is $\left(\cos(\theta),\sin(\theta)\right)$. For the angle described, what is the value of $\tan(\theta)$?