Q: If $$\tan \theta = 3$$ and $$\theta is in $$the second quadrant, what is $$\sin \theta$$?

$$\sin \theta = \frac{3}{\sqrt{10}} ($$but negative in QII, so $$\sin \theta = \frac{3}{\sqrt{10}})$$

Q: Which of the following is the correct definition of polar coordinates?

A point is defined by $$(r, \theta)$$, where $$r is $$the distance from the origin and $$\theta is $$the angle from the positive $$x-$$axis.

Q: Which of the following is the correct double angle identity for $$\cos(2\theta)$$?

$$\cos(2\theta) = \cos^2$ \theta - \sin^2$ \theta$$

Question 1

Which of the following correctly expresses the sine of a sum of two angles $$\alpha$$ and $$\beta$$ using the sum and difference identities?

Accepted Answer

$$\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$$

Question 2

A Ferris wheel has a radius of 20 meters. If a passenger is at an angle of $$	heta = \frac{\pi}{3}$$ radians from the horizontal, what is the vertical height above the center of the wheel?

Accepted Answer

$$20 \cdot \sin\left(\frac{\pi}{3}\right) = 10 \sqrt{3}$$ meters

Question 3

Which of the following is the correct value for $$\cos\left(\frac{5\pi}{6}\right)$$?

Accepted Answer

$$-\frac{\sqrt{3}}{2}$$

Question 4

Given the equation $$y = 3 \sin(2x - \frac{\pi}{4})$$, what is the amplitude, period, and phase shift?

Accepted Answer

Amplitude: $$3$$, Period: $$\pi$$, Phase shift: $$\frac{\pi}{8}$$ right

Question 5

A triangle has sides $$a = 7$$, $$b = 10$$, and angle $$C = 45^\circ. $$Using the Law of Cosines, what is the length of side $$c$$ opposite angle $$C$$?

Accepted Answer

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

Question 6

Which of the following is the correct expression for the area $$A of a $$triangle given two sides $$a$$, $$b$$ and the included angle $$C$$?

Accepted Answer

$$A = \frac{1}{2} ab \sin C$$

Question 7

If $$	an 	heta = 3$$ and $$	heta is in $$the second quadrant, what is $$\sin 	heta$$?

Accepted Answer

$$\sin 	heta = \frac{3}{\sqrt{10}} ($$but negative in QII, so $$\sin 	heta = \frac{3}{\sqrt{10}})$$

Question 8

Which of the following is the correct definition of polar coordinates?

Accepted Answer

A point is defined by $$(r, 	heta)$$, where $$r is $$the distance from the origin and $$	heta is $$the angle from the positive $$x-$$axis.

Question 9

Given the vector $$\vec{v} = \langle 4, 3 \rangle$$, what is its magnitude?

Accepted Answer

$$\sqrt{4^2$ + 3^2$} = 5$$

Question 10

Which of the following is the correct value for $$\sin(225^\circ)$$?

Accepted Answer

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

Question 11

A sound wave can be modeled by $$y = 2 \cos(5t). $$What is the period of the wave?

Accepted Answer

$$\frac{2\pi}{5}$$

Question 12

If $$\sin 	heta = \frac{1}{2}$$ and $$	heta is in $$the first quadrant, what is $$	heta in $$degrees?

Accepted Answer

$$30^\circ$$

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

Which of the following is the correct expression for the dot product of two vectors $$\vec{a} = \langle a_1$, a_2$ \rangle$$ and $$\vec{b} = \langle b_1$, b_2$ \rangle$$?

Accepted Answer

$$\vec{a} \cdot \vec{b} = a_1$ b_1$ + a_2$ b_2$

Question 15

Solve for $$x in$$ $$\sin x = 0.6$$ where $$x is in$$ $$[0, 2\pi]. $$Which of the following is correct?

Accepted Answer

$$x = \arcsin(0.6)$$ and $$x = \pi - \arcsin(0.6)$$

Question 16

Which of the following is the correct double angle identity for $$\cos(2	heta)$$?

Accepted Answer

$$\cos(2	heta) = \cos^2$ 	heta - \sin^2$ 	heta$$

Question 17

A pilot flies 100 miles north, then 100 miles east. What is the magnitude of the displacement vector?

Accepted Answer

$$\sqrt{100^2$ + 100^2$} = 100 \sqrt{2}$$ miles

Question 18

Which of the following is the correct expression for converting from polar to rectangular coordinates?

Accepted Answer

$$x = r \cos 	heta$$, $$y = r \sin 	heta$$

Question 19

If $$\cos 	heta = -\frac{1}{2}$$ and $$	heta is in$$ $$[0, 2\pi]$$, which values of $$	heta$$ satisfy this equation?

Accepted Answer

$$	heta = \frac{2\pi}{3}, \frac{4\pi}{3}$$

MAT 1560 Trigonometry Syllabus and Study Guide Overview

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