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.

Answer the following:

Fill out this table. It will probably be helpful in the future.

$t$	$\tan(t)$
$0$
$\frac{\pi}{6}$
$\frac{\pi}{4}$
$\frac{\pi}{3}$
$\frac{\pi}{2}$

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. $t = -\frac{11\pi}{6}$
Why is $\tan\left(\frac{\pi}{2}\right)$ undefined?
$P\left(-\dfrac{3}{5}, -\dfrac{4}{5}\right)$ is a terminal point associated with some $t \in \mathbb{R}$. Find $\sin(t), \cos(t)$ and $\tan(t)$.
Let's say $\cos(t) = -\frac{7}{25}$ and the terminal point of $t$ is in Quadrant III. What is $\sin(t)$ and $\tan(t)$?
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. $y = 5\sec\left(\pi x + \frac{\pi}{3}\right)$
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?
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?
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.
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.
Out of the following, which ones are true? 🤔 Justify your answer with the domain of the proper function.

$\sin^{-1}\left(\sin\left(\dfrac{\pi}{3}\right)\right) = \dfrac{\pi}{3}$
$\sin^{-1}\left(\sin\left(-\dfrac{\pi}{4}\right)\right) = -\dfrac{\pi}{4}$
$\sin^{-1}\left(\sin\left(-\dfrac{2\pi}{3}\right)\right) = -\dfrac{2\pi}{3}$
$\cos^{-1}\left(\cos\left(-\dfrac{2\pi}{3}\right)\right) = -\dfrac{2\pi}{3}$
$\cos\left(\cos^{-1}\left(\dfrac{1}{2}\right)\right) = \dfrac{1}{2}$
$\tan^{-1}\left(\tan\left(-\dfrac{2\pi}{3}\right)\right) = -\dfrac{2\pi}{3}$
$\tan\left(\tan^{-1}100000000000\right) = 100000000000$

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. $\tan\left(\sin^{-1}\frac{\sqrt{2}}{2}\right)$
Suppose $\sin t = -\frac{4}{5}$ and the terminal points $P$ is in Quadrant IV. What is $\cos t$?
Are the following functions even, odd, or neither?
1. $f(x) = x^2 \sin x$
2. $f(x) = \sin(x)\cos(x)$
If $\sin t > 0$ and $\cos t < 0$, what quadrant is the terminal point in?
If $\cos t < 0$ and $\cot t < 0$, what quadrant is the terminal point in?
Is the function $f(t) = \sin t \cos t$ negative or positive when $t$ is in Quadrant IV? How about Quadrant II?
Is the function $f(t) = \frac{\tan t}{\cos t}$ negative or positive when $t$ is in Quadrant III? How about Quadrant I?
What is the domain of the function $f(t) = \csc t \sec t$?