Q: Which of the following is the correct double-angle formula for $$\cos 2\theta$$?

$$\cos 2\theta = 2\cos^2$ \theta - 1$$

Question 1

A Ferris wheel with a radius of 20 meters completes one full revolution every 2 minutes. What is the angular speed of the Ferris wheel in radians per minute?

Accepted Answer

$$\frac{\pi}{1}$$

Question 2

Convert $$135^{\circ} to $$radians.

Accepted Answer

$$\frac{3\pi}{4}$$

Question 3

Which of the following is the correct expression for the sine of the sum of two angles $$A$$ and $$B$$?

Accepted Answer

$$\sin(A+B) = \sin A \cos B + \cos A \sin B$$

Question 4

Given $$\vec{v} = 3\mathbf{i} + 4\mathbf{j}$$, what is the magnitude of $$\vec{v}$$?

Accepted Answer

$$5$$

Question 5

A ship sails 8 km east and then 6 km north. What is the magnitude of the resultant displacement vector?

Accepted Answer

$$10 km$$

Question 6

Which of the following is an even trigonometric function?

Accepted Answer

$$\cos(x)$$

Question 7

Solve for $$x in$$ $$2\sin^2$ x - 1 = 0$$ for $$0 \leq x \u003c 2\pi.$$

Accepted Answer

$$x = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}$$

Question 8

Given $$	riangle ABC$$ with $$a = 7$$, $$b = 10$$, and $$C = 45$$^{\circ}$$, use the Law of Cosines to find c$.

Accepted Answer

$$c = \sqrt{7^2$ + 10^2$ - 2 \cdot 7 \cdot 10 \cdot \cos 45$$^{\circ}$$}$$

Question 9

Which of the following is the correct double-angle formula for $$\cos 2	heta$$?

Accepted Answer

$$\cos 2	heta = 2\cos^2$ 	heta - 1$$

Question 10

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

Accepted Answer

$$-\frac{4}{5}$$

Question 11

Which of the following is the correct expression for the dot product of $$\vec{a} = a_1$\mathbf{i} + a_2$\mathbf{j}$$ and $$\vec{b} = b_1$\mathbf{i} + b_2$\mathbf{j}$$?

Accepted Answer

$$a_1b_1 + a_2b_2$$

Question 12

A sound wave can be modeled by $$y = 0.5 \sin(440\pi t)$$, where $$t is in $$seconds. What is the period of this wave?

Accepted Answer

$$\frac{1}{220}$$ seconds

Question 13

Which of the following is the correct formula for the Law of Sines?

Accepted Answer

$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$

Question 14

If $$	an \alpha = 2$$ and $$\alpha is in $$Quadrant III, what is $$\sin \alpha$$?

Accepted Answer

$$-\frac{2}{\sqrt{5}}$$

Question 15

Which of the following is the correct power-reducing formula for $$\sin^2$ x$$?

Accepted Answer

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

Question 16

A vector has magnitude $$10$$ and direction $$60^{\circ}$$ from the positive $$x-$$axis. What are its rectangular components?

Accepted Answer

$$5\mathbf{i} + 5\sqrt{3}\mathbf{j}$$

Question 17

Which of the following is the correct formula for the arc length $$s of a $$circle of radius $$r$$ subtended by a central angle $$	heta (in $$radians)?

Accepted Answer

$$s = r	heta$$

Question 18

If $$\cos$$^{-1}$$(x) = \frac{\pi}{3}$$, what is $$x$$?

Accepted Answer

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

Question 19

Which of the following is the correct sum-to-product formula for $$\sin A + \sin B$$?

Accepted Answer

$$2\sin\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right)$$

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