# Homework 4

Directions: Turn in your answers in class on a physical piece of paper. Staple multiple sheets together. Feel free to use Desmos for graphing.

Before you start, make sure you know how to find reference numbers for any $t \in \mathbb{R}$. Refresh up on Section 5.1, especially if you did not do well on the last problem on the midterm.

1. Fill out this table. It will probably be helpful in the future.
$t$ $\tan(t)$
$0$
$\frac{\pi}{6}$
$\frac{\pi}{4}$
$\frac{\pi}{3}$
$\frac{\pi}{2}$
2. Using the three step process, find the six trigonmetric functions of the following values of $t$:
1. $t = \frac{10\pi}{3}$
2. $t = \frac{7\pi}{6}$
3. $t = -\frac{2\pi}{3}$
4. $t = -\frac{11\pi}{6}$
3. Why is $\tan\left(\frac{\pi}{2}\right)$ undefined?
4. $P\left(-\dfrac{3}{5}, -\dfrac{4}{5}\right)$ is a terminal point associated with some $t \in \mathbb{R}$. Find $\sin(t), \cos(t)$ and $\tan(t)$.
5. Let's say $\cos(t) = -\frac{7}{25}$ and the terminal point of $t$ is in Quadrant III. What is $\sin(t)$ and $\tan(t)$?
6. State each relevant transformation, amplitude, and period (amplitude only for $\sin, \cos$). Graph one complete period by hand. Be sure to show in the graph the endpoints of one period.
1. $y = 1 + \cos \pi x$
2. $y = \dfrac{1}{2} - \dfrac{1}{2}\cos\left(2x - \dfrac{\pi}{3}\right)$
3. $y = \dfrac{1}{2} - \dfrac{1}{2}\cos\left(2\left(x - \dfrac{\pi}{3}\right)\right)$
4. $y = \sin(\pi + 2x)$
5. $y = 1 + \tan\left(\pi x\right)$
6. $y = 4 \tan (3x - 2\pi)$
7. $y = 5\sec\left(\pi x + \frac{\pi}{3}\right)$
7. Suppose $f(x) = x \qquad\qquad g(x) = \sin(x)$ Graph $f(x) + g(x)$ in Desmos. Based on what you know about graphical addition, why does the graph $f(x) + g(x)$ look like the way it does?
8. Because $f(x) = \sin(x)$ is not one to one, what is the interval of values we restrict the domain of $f(x)$ to in order for $f(x)$ to have an inverse?
9. Inverse function property in 2.8 tells us that if $f(x) = \sin(x)$ and $g(x) = \sin^{-1}(x)$, then $(f\circ g)(x) = x$ is valid. For what values of $x$ is $(f \circ g)(x) = x$ true? Moreover, explain the exact reason why these specific values of $x$ are valid.
10. Inverse function property in 2.8 tells us that $\cos^{-1}(\cos(x)) = x$. For what values of $x$ is this true? Moreover, explain the exact reason why these specific values of $x$ are valid.
11. Out of the following, which ones are true? 🤔 Justify your answer with the domain of the proper function.
1. $\sin^{-1}\left(\sin\left(\dfrac{\pi}{3}\right)\right) = \dfrac{\pi}{3}$
2. $\sin^{-1}\left(\sin\left(-\dfrac{\pi}{4}\right)\right) = -\dfrac{\pi}{4}$
3. $\sin^{-1}\left(\sin\left(-\dfrac{2\pi}{3}\right)\right) = -\dfrac{2\pi}{3}$
4. $\cos^{-1}\left(\cos\left(-\dfrac{2\pi}{3}\right)\right) = -\dfrac{2\pi}{3}$
5. $\cos\left(\cos^{-1}\left(\dfrac{1}{2}\right)\right) = \dfrac{1}{2}$
6. $\tan^{-1}\left(\tan\left(-\dfrac{2\pi}{3}\right)\right) = -\dfrac{2\pi}{3}$
7. $\tan\left(\tan^{-1}100000000000\right) = 100000000000$
12. Evaluate the following:
1. $\sin^{-1}\left(\dfrac{\sqrt{3}}{2}\right)$
2. $\sin^{-1}\left(\dfrac{-1}{2}\right)$
3. $\cos^{-1}\left(-\dfrac{\sqrt{2}}{2}\right)$
4. $\cos^{-1}(2)$
5. $\tan^{-1}(0)$
6. $\tan^{-1}(\sqrt{3})$
7. $\tan\left(\sin^{-1}\frac{\sqrt{2}}{2}\right)$
13. Suppose $\sin t = -\frac{4}{5}$ and the terminal points $P$ is in Quadrant IV. What is $\cos t$?
14. Are the following functions even, odd, or neither?
1. $f(x) = x^2 \sin x$
2. $f(x) = \sin(x)\cos(x)$
15. If $\sin t > 0$ and $\cos t < 0$, what quadrant is the terminal point in?
16. If $\cos t < 0$ and $\cot t < 0$, what quadrant is the terminal point in?
17. Is the function $f(t) = \sin t \cos t$ negative or positive when $t$ is in Quadrant IV? How about Quadrant II?
18. Is the function $f(t) = \frac{\tan t}{\cos t}$ negative or positive when $t$ is in Quadrant III? How about Quadrant I?
19. What is the domain of the function $f(t) = \csc t \sec t$?