Show each step of your work and fully simplify each expression.
Turn in your answers in class on a physical piece of paper.
Staple multiple sheets together.
Feel free to use Desmos for graphing.
Answer the following:
Is the point $P\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$ on the unit circle? Show using calculations.
What is the reference number $\bar{t}$? What is the range of values $\bar{t}$ is allowed to be?
Find the following terminal points $P$ associated with the following $t$ values:
$t = 0$
$t = \frac{3\pi}{2}$
$t = \frac{2\pi}{3}$
$t = \frac{5\pi}{6}$
$t = -\frac{2\pi}{3}$
$t = -\frac{9\pi}{4}$
$t = -\frac{11\pi}{6}$
$t = 1000\pi + \frac{2\pi}{3}$
$t = -83\pi + \frac{\pi}{3}$
$t = \pi + 2\pi - 3\pi + \frac{4\pi}{3}$
$t = \dfrac{81\pi}{6}$
Find the six trigonometric functions of $t = \dfrac{2\pi}{3}$.
Find the six trigonometric functions of $t = -\dfrac{5\pi}{6}$.
Find the six trigonometric functions of $t = \dfrac{43\pi}{6}$.
Using the terminal point associated with $t = \dfrac{\pi}{2}$, explain why $\tan\left(\dfrac{\pi}{2}\right)$ does not exist.
A function usually has the notation $y = f(x)$, meaning you plug in $x$ into $f$ and the result is called $y$.
Let $x \in \mathbb{R}$ and $P(a, b)$ be the associated terminal point. What should the value of $y$ in the equation \[y = \cos x\] be?
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?
Consider the function \[f(t) = \sin (t) \cos (t)\]
Determine if $f(t)$ is positive or negative when the terminal point $P$ associated with $t$ is in Quadrant I, II, III, and IV.
Hint: Use the definition of the trig function + their coordinate signs.
Consider the function \[f(t) = \sec^{15}(t) \cdot \left[1 - \tan(t)\right]^6\]
Determine if $f(t)$ is positive or negative when the terminal point $P$ associated with $t$ is in Quadrant I, II, III, and IV.
Suppose $t \in \mathbb{R}$.
What is the maximum value $\sin (t)$ could be? How about the minimum value?
Answer the two questions above for $\cos (t)$.
Fill out this table. It will probably be helpful in the future.
$t$
$\tan(t)$
$0$
$\frac{\pi}{6}$
$\frac{\pi}{4}$
$\frac{\pi}{3}$
$\frac{\pi}{2}$
$P\left(-\dfrac{3}{5}, -\dfrac{4}{5}\right)$ is a terminal point associated with some $t \in \mathbb{R}$. Determine the values of $\sin(t), \cos(t)$ and $\tan(t)$.
Suppose a function $f(x)$ has period $23$. Write the mathematical formula describing this behavior.
What is the period of the following function?
Suppose $f(x) = \sqrt{x}$. We want to vertically shift $f(x)$ upwards 3 units and left four units to make a new function $g(x)$. What is the formula of $g(x)$?
Starting from a base function $f(x)$, I want to create a function $g(x)$ where I apply the following transformations to $f(x)$ in the following order:
Is $g(x)$ horizontally shifted four units to the left from $f(x)$? Why or why not?
Is $h(x)$ horizontally shifted four units to the left from $f(x)$? Why or why not?
Is $i(x)$ horizontally shifted four units to the left from $f(x)$? Why or why not?
Graph one period of $f(x) = \cos x$ by hand.
For each of the following functions:
1. State each relevant transformation, amplitude, and period. 2. Graph one complete period by hand. Be sure to show in the graph the endpoints of one period.
(skip this) $y = 5\sin\left(\pi x + \dfrac{\pi^2}{3}\right)$
(skip this) $y = -5 + 4\cos(\pi - x)$
(skip this) For each of the following functions:
1. State each relevant transformation. 2. Graph one complete period by hand. Be sure to show in the graph where one period beings and ends.
$y = 5\sec x$
$f(x) = 2\csc \left(\pi x - \dfrac{\pi^2}{2}\right)$