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 $t$:
 $t = \frac{10\pi}{3}$
 $t = \frac{7\pi}{6}$
 $t = \frac{2\pi}{3}$
 $t = \frac{11\pi}{6}$
 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.
 $y = 1 + \cos \pi x$
 $y = \dfrac{1}{2}  \dfrac{1}{2}\cos\left(2x  \dfrac{\pi}{3}\right)$
 $y = \dfrac{1}{2}  \dfrac{1}{2}\cos\left(2\left(x  \dfrac{\pi}{3}\right)\right)$
 $y = \sin(\pi + 2x)$
 $y = 1 + \tan\left(\pi x\right)$
 $y = 4 \tan (3x  2\pi)$
 $y = 5\sec\left(\pi x + \frac{\pi}{3}\right)$
 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? ðŸ¤” 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:
 $\sin^{1}\left(\dfrac{\sqrt{3}}{2}\right)$
 $\sin^{1}\left(\dfrac{1}{2}\right)$
 $\cos^{1}\left(\dfrac{\sqrt{2}}{2}\right)$
 $\cos^{1}(2)$
 $\tan^{1}(0)$
 $\tan^{1}(\sqrt{3})$
 $\tan\left(\sin^{1}\frac{\sqrt{2}}{2}\right)$

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?

$f(x) = x^2 \sin x$

$f(x) = \sin(x)\cos(x)$

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 $f(t) = \csc t \sec t$?