Question 1

Prove the following trigonometric identity: $$\csc 2x + \cot 2x = \cot x.$$

Accepted Answer

By expressing $$\csc 2x$$ and $$\cot 2x in $$terms of $$x$$ and simplifying, the identity holds.

Question 2

Simplify $$\cos \left( x + \frac{\pi}{2} \right).$$

Accepted Answer

$$-\sin x$$

Question 3

Find the exact value of $$\sin \left( -\frac{7\pi}{12} \right).$$

Accepted Answer

$$\frac{\sqrt{6} - \sqrt{2}}{4}$$

Question 4

Given $$\sin x = \frac{3}{8}$$ and $$0 \u003c x \u003c \frac{\pi}{2}$$, find the exact value of $$\sin 2x.$$

Accepted Answer

$$\frac{3\sqrt{55}}{32}$$

Question 5

Given triangle ABC with $$a = 9$$, $$b = 7$$, $$c = 5$$, which law would you use to solve for the angles?

Accepted Answer

Law of Cosines

Question 6

If $$\sin x = \frac{3}{8}$$ and $$0 \u003c x \u003c \frac{\pi}{2}$$, what is $$\cos x$$?

Accepted Answer

$$\frac{\sqrt{55}}{8}$$

Question 7

Which of the following is the correct formula for $$\sin 2x in $$terms of $$\sin x$$ and $$\cos x$$?

Accepted Answer

$$2 \sin x \cos x$$

Question 8

If $$\cos x = \frac{\sqrt{55}}{8}$$ and $$\sin x = \frac{3}{8}$$, what is $$	an x$$?

Accepted Answer

$$\frac{3}{\sqrt{55}}$$

Question 9

Which of the following is the correct value for $$\cos \left( x + \frac{\pi}{2} \right)$$?

Accepted Answer

$$-\sin x$$

Question 10

Given triangle ABC with sides $$a = 9$$, $$b = 7$$, $$c = 5$$, what is the measure of angle $$A ($$opposite side $$a$$)?

Accepted Answer

$$\arccos \left( \frac{b^2$ + c^2$ - a^2$}{2bc} \right)$$

Question 11

If $$\sin x = \frac{3}{8}$$ and $$0 \u003c x \u003c \frac{\pi}{2}$$, what is $$\sin^2$ x + \cos^2$ x$$?

Accepted Answer

$$1$$

Question 12

Which of the following is the correct value for $$\sin \left( -\frac{7\pi}{12} \right)$$?

Accepted Answer

$$\frac{\sqrt{6} - \sqrt{2}}{4}$$

Question 13

If $$\sin x = \frac{3}{8}$$, what is $$\sin 2x$$?

Accepted Answer

$$2 \cdot \frac{3}{8} \cdot \frac{\sqrt{55}}{8}$$

Question 14

Which trigonometric identity is used to simplify $$\cos \left( x + \frac{\pi}{2} \right)$$?

Accepted Answer

Angle addition formula

Question 15

Given $$\sin x = \frac{3}{8}$$ and $$0 \u003c x \u003c \frac{\pi}{2}$$, which quadrant is $$x in$$?

Accepted Answer

First quadrant

Trigonometric Identities, Simplification, and Triangle Solving

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