Q: Prove the identity: $\tan x + \cot x = \frac{\sin^2 x + \cos^2 x}{\sin x \cos x}$.

True, because $\sin^2 x + \cos^2 x = 1$ and $\tan x + \cot x = \frac{1}{\sin x \cos x}$

Question 1

What is the exact radian value of $\arccos(\frac{1}{2})$?

Accepted Answer

$\frac{\pi}{3}$

Question 2

Evaluate $\arcsin(-\frac{\sqrt{3}}{2})$.

Accepted Answer

$-\frac{\pi}{3}$

Question 3

Find the exact value in degrees of $\arccos(0)$.

Accepted Answer

$90^\circ$

Question 4

If $\sin 	heta = \frac{3}{5}$ and $	heta$ is in quadrant II, what is $\cos 	heta$?

Accepted Answer

$-\frac{4}{5}$

Question 5

Write $	an(\arccos(x))$ in terms of $x$ only.

Accepted Answer

$\frac{\sqrt{1-x^2}}{x}$

Question 6

Simplify $\frac{\sin^2 x}{1-\cos x}$.

Accepted Answer

$1+\cos x$

Question 7

Determine whether the function $f(x) = x^3 - x$ is even, odd, or neither.

Accepted Answer

Odd

Question 8

Factor $\sin^2 x - \cos^2 x$.

Accepted Answer

$(\sin x - \cos x)(\sin x + \cos x)$

Question 9

Prove the identity: $	an x + \cot x = \frac{\sin^2 x + \cos^2 x}{\sin x \cos x}$.

Accepted Answer

True, because $\sin^2 x + \cos^2 x = 1$ and $	an x + \cot x = \frac{1}{\sin x \cos x}$

Question 10

Solve the right triangle where $a = 7$, $c = 14$. What is $b$?

Accepted Answer

$b = 7\sqrt{3}$

Question 11

Convert $3.14770$ radians to degrees (rounded to two decimal places).

Accepted Answer

$180.43^\circ$

Question 12

Find the value of $\sin(2.5464)$ (rounded to four decimal places).

Accepted Answer

$0.5649$

Question 13

A ship at sea spots a lighthouse at a $5^\circ$ angle of elevation. If the lighthouse is $150$ feet tall, how far is the ship from the base of the lighthouse (to the nearest foot)?

Accepted Answer

$1718$ feet

Question 14

If $	an 	heta = \frac{3}{4}$ and $	heta$ is in quadrant III, what is $\sin 	heta$?

Accepted Answer

$-\frac{3}{5}$

Question 15

Which of the following is the period of the function $f(x) = 2\sin(3x)$?

Accepted Answer

$\frac{2\pi}{3}$

Question 16

Given $\sin x = \frac{1}{2}$ and $x$ is in $[0, 2\pi]$, what are all possible values of $x$?

Accepted Answer

$\frac{\pi}{6}, \frac{5\pi}{6}$

Question 17

If $\cos(	heta) = -0.6$ and $	heta$ is in quadrant III, what is $\sin(	heta)$?

Accepted Answer

$-0.8$

Question 18

Which of the following is an identity?

Accepted Answer

$\sin^2 x + \cos^2 x = 1$

Question 19

A 40-foot vertical cliff is observed from a boat at a $12^\circ$ angle of elevation. How far is the boat from the base of the cliff (to the nearest foot)?

Accepted Answer

$188$ feet

Question 20

If $\sin(\alpha) = 0.6$ and $\alpha$ is in quadrant II, what is $\cos(\alpha)$?

Accepted Answer

$-0.8$

Precalculus Review: Trigonometric Functions, Identities, and Applications

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