Q: Which of the following statements is TRUE?

$$\cos(\cos$$^{-1}$$(x)) = x$$ \text{ for } x \in [-1, 1]$

Q: Simplify $$\tan(x + \frac{\pi}{4}).$$

$$\frac{\tan(x) + 1}{1 - \tan(x)}$$

Question 1

What is the range of the function $$f(x) = \sin$$^{-1}$$(x)$$?

Accepted Answer

$$\left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]$$

Question 2

Which of the following statements is TRUE?

Accepted Answer

$$\cos(\cos$$^{-1}$$(x)) = x$$ 	ext{ for } x \in [-1, 1]$

Question 3

Find the exact value of $$\cos$$^{-1}$$(\cos(\frac{9\pi}{8})).$$

Accepted Answer

$$\frac{7\pi}{8}$$

Question 4

Given a triangle for which one side and three angles are known, which law is used to solve for the second side?

Accepted Answer

Law of Sines

Question 5

Rewrite $$\cos(	an$$^{-1}$$(x)) as an $$algebraic expression in terms of $$x.$$

Accepted Answer

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

Question 6

Simplify $$	an(x + \frac{\pi}{4}).$$

Accepted Answer

$$\frac{	an(x) + 1}{1 - 	an(x)}$$

Question 7

The expression $$\sin(x) \cos(50^\circ) - \cos(x) \sin(50^\circ) is $$equivalent to:

Accepted Answer

$$\sin(x - 50^\circ)$$

Question 8

Write $$\cos(2 \cos$$^{-1}$$(4x)) in $$terms of $$x$$ only.

Accepted Answer

$$32x^2 - 1$

Question 9

A car travels along a straight road for 30 minutes, then turns $$20^\circ to $$the left of its original course and continues for 15 minutes. If the car travels at a constant speed of 10 mph, how far is the car from the starting point?

Accepted Answer

$$\sqrt{(5)^2$ + (2.5)^2$ - 2 \cdot 5 \cdot 2.5 \cos(20^\circ)}$$ miles

Question 10

Solve the trigonometric equation: $$2\cos^2$(	heta) - 5\cos(	heta) - 3 = 0. $$Which of the following is a solution for $$\cos(	heta)$$?

Accepted Answer

$$-\frac{1}{2}$$

Question 11

Find the exact value of $$\cos(\csc$$^{-1}$$(\frac{5}{2})).$$

Accepted Answer

$$\frac{\sqrt{21}}{5}$$

Question 12

Simplify $$\left( \frac{1 - \cos(3x)}{\sin(3x)} \right) - \left( \frac{\sin(3x)}{1 + \cos(3x)} \right).$$

Accepted Answer

$$\frac{2 \sin(3x)}{1 + \cos(3x)}$$

Question 13

A triangle has sides $$a = 10$$, $$b = 7$$, and angle $$\angle A = 140^\circ. $$Find angle $$\angle B.$$

Accepted Answer

no solution

Question 14

Find $$\sin(2x) if$$ $$\sin(x) = \frac{3}{7}$$ and $$x is in $$quadrant II.

Accepted Answer

$$-\frac{12\sqrt{10}}{49}$$

Question 15

If $$\sec(x) = \frac{7}{2}$$ and $$270^\circ \u003c x \u003c 360^\circ$$, what is the exact value of $$\cos\left(\frac{x}{2}\right)$$?

Accepted Answer

$$-\frac{3}{\sqrt{14}}$$

Question 16

Which formula expresses $$\sin(x + y) in $$terms of $$\sin(x)$$, $$\cos(x)$$, $$\sin(y)$$, and $$\cos(y)$$?

Accepted Answer

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

Question 17

Solve for $$x$$: $$	an(5x) - \sqrt{3} = 0.$$

Accepted Answer

$$\frac{\pi}{15} + k\frac{\pi}{5}$$

Question 18

Express $$3\csc(x)	an(x) + \sec(x) in $$terms of $$\cos(x)$$ only.

Accepted Answer

$$\frac{3}{\cos(x)}\cdot\frac{\sin(x)}{\sin(x)} + \frac{1}{\cos(x)} = \frac{3\sin(x)}{\cos^2$(x)} + \frac{1}{\cos(x)}$$

Trigonometry Midterm 2 Study Guidance

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