Q: A surveyor needs to find the height of a building. Standing 50 meters from the base, she measures the angle of elevation to the top as $$30^\circ. $$What is the height of the building?

$$50 \times \tan 30^\circ \approx 28.87$$ meters

Q: Which of the following is a fundamental trigonometric identity?

$$\sin^2$ \theta + \cos^2$ \theta = 1$$

Q: A pilot is flying at an altitude of 3000 meters and spots an airport at an angle of depression of $$12^\circ. $$What is the horizontal distance to the airport?

$$\dfrac{3000}{\tan 12^\circ} \approx 14,107$$ meters

Q: Which trigonometric identity is used to simplify $$\sin(2\theta)$$?

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

Q: 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 $$C$$?

$$c^2$$ = $$7^2$$ + $$10^2 - 2$$ \times 7 \times 10 \times \cos 45^\circ$$

Q: Which of the following is NOT a valid trigonometric identity?

$$\sin^2$ \theta + \cos^2$ \theta = 2$$

Q: A ship sails 100 km east, then turns $$60^\circ$$ north of east and sails another 150 km. What is the straight-line distance from the starting point to the final position?

$$\sqrt{100^2$ + 150^2$ - 2 \times 100 \times 150 \times \cos 120^\circ}$$

Question 1

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

Accepted Answer

The ratio of the opposite side to the hypotenuse.

Question 2

A surveyor needs to find the height of a building. Standing 50 meters from the base, she measures the angle of elevation to the top as $$30^\circ. $$What is the height of the building?

Accepted Answer

$$50 	imes 	an 30^\circ \approx 28.87$$ meters

Question 3

Which of the following is a fundamental trigonometric identity?

Accepted Answer

$$\sin^2$ 	heta + \cos^2$ 	heta = 1$$

Question 4

If $$\sin 	heta = 0.6$$ and $$	heta is in $$the first quadrant, what is $$\cos 	heta$$?

Accepted Answer

$$0.8$$

Question 5

Which equation correctly expresses the Law of Sines?

Accepted Answer

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

Question 6

A pilot is flying at an altitude of 3000 meters and spots an airport at an angle of depression of $$12^\circ. $$What is the horizontal distance to the airport?

Accepted Answer

$$\dfrac{3000}{	an 12^\circ} \approx 14,107$$ meters

Question 7

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

Accepted Answer

$$a^2$$ = $$b^2$$ + $$c^2 - 2bc$$ \cos A$$

Question 8

Which trigonometric identity is used to simplify $$\sin(2	heta)$$?

Accepted Answer

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

Question 9

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

$$c^2$$ = $$7^2$$ + $$10^2 - 2$$ 	imes 7 	imes 10 	imes \cos 45^\circ$$

Question 10

Which of the following is the correct expression for the period of the function $$f(x) = \sin(kx)$$?

Accepted Answer

$$\dfrac{2\pi}{k}$$

Question 11

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

Accepted Answer

$$-\dfrac{3}{\sqrt{10}}$$

Question 12

Which of the following is NOT a valid trigonometric identity?

Accepted Answer

$$\sin^2$ 	heta + \cos^2$ 	heta = 2$$

Question 13

A ship sails 100 km east, then turns $$60^\circ$$ north of east and sails another 150 km. What is the straight-line distance from the starting point to the final position?

Accepted Answer

$$\sqrt{100^2$ + 150^2$ - 2 	imes 100 	imes 150 	imes \cos 120^\circ}$$

Question 14

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

Accepted Answer

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

Question 15

A Ferris wheel has a radius of 20 meters. If a passenger is at an angle of $$135^\circ$$ from the horizontal, what is the vertical height above the center?

Accepted Answer

$$20 	imes \sin 135^\circ \approx 14.14$$ meters

Question 16

Which of the following is the correct expression for the amplitude of $$f(x) = 5 \sin(x) - 3$$?

Accepted Answer

$$5$$

Question 17

If $$\cos 	heta = -0.5$$ and $$	heta is in $$the third quadrant, what is $$	heta$$?

Accepted Answer

$$240^\circ$$

Question 18

Which of the following is the correct formula for converting degrees to radians?

Accepted Answer

$$	ext{radians} = \dfrac{\pi}{180} 	imes 	ext{degrees}$$

Question 19

A sound wave can be modeled by $$y = 3 \sin(440 \pi t)$$, where $$t is $$time in seconds. What is the frequency of the wave?

Accepted Answer

$$220 Hz$$

MAT147 – College Trigonometry Syllabus and Study Guide

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