2.3 The Sine and Cosine Functions (continued)

Using computing technology

Summary

The $x$ coordinate of the point on the unit circle that lies in the third quadrant and whose $y$-coordinate is $y = -\frac{3}{4}$.

The $y$-coordinate of the point on the unit circle generated by a central angle opening counterclockwise with one side on the positive $x$-axis that measures $t = 2$ radians.

The $x$-coordinate of the point on the unit circle generated by a central angle with one side on the positive $x$-axis that measures $t = -3.05$ radians. (With the negative radian measure, we view the angle as opening clockwise from its initial side on the positive $x$-axis.)

The value of $\cos(t)$ where $t$ is an angle in Quadrant II that satisfies $\sin(t) = \frac{1}{2}$.

The value of $\sin(t)$ where $t$ is an angle in Quadrant III for which $\cos(t) = -0.7$.

The average rate of change of $f(t) = \sin(t)$ on the intervals $[0.1,0.2]$ and $[0.8,0.9]$.

The average rate of change of $g(t) = \cos(t)$ on the intervals $[0.1,0.2]$ and $[0.8,0.9]$.

Without using a computational device, determine the exact value of each of the following quantities.

-$\sin(-\frac{11\pi}{4})$-$\cos(\frac{29\pi}{6})$-$\cos(\frac{17\pi}{3})$-$\sin(47\pi)$-$\cos(-113\pi)$-$t$ in quadrant III such that $\cos(t) = -\frac{\sqrt{3}}{2}$-$t$ in quadrant IV such that $\sin(t) = -\frac{\sqrt{3}}{2}$

We now know three different identities involving the sine and cosine functions: $\sin(t+\frac{\pi}{2}) = \cos(t)$, $\cos(t-\frac{\pi}{2}) = \sin(t)$, and $\cos^2(t) + \sin^2(t) = 1$. Following are several proposed identities. For each, your task is to decide whether the identity is true or false. If true, give a convincing argument for why it is true; if false, give an example of a $t$-value for which the equation fails to hold.

-$\cos(t + 2\pi) = \cos(t)$-$\sin(t-\pi) = -\sin(t)$-$\cos(t - \frac{3\pi}{2}) = \sin(t)$-$\sin^2(t) = 1 - \cos^2(t)$-$\sin(t) + \cos(t) = 1$-$\sin(t) + \sin(\frac{\pi}{2}) = \cos(t)$