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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