Q: Which of the following is the correct definition of the sine function for an acute angle $$\theta in a $$right triangle?

$$\sin \theta = \dfrac{\text{opposite}}{\text{hypotenuse}}$$

Q: Which of the following is the correct double-angle identity for cosine?

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

Q: A ship sails 8 km east, then 6 km north. What is the straight-line distance from the starting point to the ship's final position, and what is the bearing from the starting point?

$$10 km$$, $$\tan$$^{-1}$$(\frac{6}{8}) \approx 36.87^\circ$$ north of east

Question 1

Which of the following is the correct definition of the sine function for an acute angle $$	heta in a $$right triangle?

Accepted Answer

$$\sin 	heta = \dfrac{	ext{opposite}}{	ext{hypotenuse}}$$

Question 2

A ladder leans against a wall, forming a $$65^\circ$$ angle with the ground. If the foot of the ladder is 4 meters from the wall, how long is the ladder? (Round to two decimal places.)

Accepted Answer

$$4.41 / \cos 65^\circ \approx 9.48$$ meters

Question 3

Which of the following is an identity for $$\sin(\alpha + \beta)$$?

Accepted Answer

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

Question 4

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

Accepted Answer

$$\dfrac{3}{\sqrt{10}} ($$positive in QII)

Question 5

A Ferris wheel has a radius of 20 meters. If a passenger is at a position that forms a $$120^\circ$$ central angle from the ground, how high above the ground is the passenger (assuming the center is 20 meters above the ground)?

Accepted Answer

$$20 + 20 \sin 120^\circ \approx 37.32$$ meters

Question 6

Which of the following is the correct double-angle identity for cosine?

Accepted Answer

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

Question 7

If $$\sin 	heta = \dfrac{5}{13}$$ and $$	heta is in $$the first quadrant, what is $$\cos 	heta$$?

Accepted Answer

$$\dfrac{12}{13}$$

Question 8

A ship sails 8 km east, then 6 km north. What is the straight-line distance from the starting point to the ship's final position, and what is the bearing from the starting point?

Accepted Answer

$$10 km$$, $$	an$$^{-1}$$(\frac{6}{8}) \approx 36.87^\circ$$ north of east

Question 9

Which of the following is the correct form of the Law of Sines?

Accepted Answer

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

Question 10

Given $$\cos 	heta = -\dfrac{3}{5}$$ and $$	heta is in $$the third quadrant, what is $$\sin 	heta$$?

Accepted Answer

$$-\dfrac{4}{5}$$

Question 11

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

Accepted Answer

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

Question 12

A point $$P on $$the unit circle corresponds to an angle $$	heta = 225^\circ. $$What are the coordinates of $$P$$?

Accepted Answer

$$\left( -\dfrac{\sqrt{2}}{2}, -\dfrac{\sqrt{2}}{2} \right)$$

Question 13

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

Accepted Answer

$$r = \sqrt{x^2$ + y^2$}$$, $$	heta = 	an$$^{-1}$$ \left( \dfrac{y}{x} ight)$$

Question 14

If $$\sin x = \dfrac{1}{2}$$ and $$0 \leq x \u003c 2\pi$$, what are all possible values of $$x$$?

Accepted Answer

$$x = \dfrac{\pi}{6}, \dfrac{5\pi}{6}$$

Question 15

A rotating radar antenna completes 3 full revolutions per minute. What is its angular speed in radians per second?

Accepted Answer

$$\dfrac{3 	imes 2\pi}{60} = \dfrac{\pi}{10}$$ radians/second

Question 16

Which of the following is the correct form of the Law of Cosines?

Accepted Answer

$$c^2$$ = $$a^2$$ + $$b^2 - 2ab$$ \cos C$$

Question 17

Which of the following is the correct expression for $$	an(\alpha - \beta)$$?

Accepted Answer

$$\dfrac{	an \alpha - 	an \beta}{1 + 	an \alpha 	an \beta}$$

Question 18

If $$z = 2 	ext{cis} \dfrac{\pi}{3}$$ and $$w = 3 	ext{cis} \dfrac{\pi}{6}$$, what is $$z \cdot w in $$polar form?

Accepted Answer

$$6 	ext{cis} \left( \dfrac{\pi}{3} + \dfrac{\pi}{6} ight ) = 6 	ext{cis} \dfrac{\pi}{2}$$

Question 19

Which of the following is DeMoivre's Theorem?

Accepted Answer

$$(r 	ext{cis} 	heta)^n = r^n 	ext{cis}(n	heta)$$

Trigonometry: Core Concepts, Functions, and Applications – Study Notes

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