Q: A surveyor measures the angle of elevation from point A to the top of a building as $$35^\\circ. If $$point A is 50 meters from the base of the building, what is the height of the building (to the nearest meter)?

$$\tan 35^\circ = \frac{h}{50}$$, so $$h = 50 \times \tan 35^\circ \approx 35$$ meters

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

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

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

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

Q: Given $$\sin \theta = 0.6$$ and $$\theta is an $$acute angle, what is $$\cos \theta$$?

$$\cos \theta = \sqrt{1 - (0.6)^2$} = 0.8$$

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

$$\sin \theta = \frac{3}{\sqrt{1 + 3^2$}} = \frac{3}{\sqrt{10}}$$, but since $$\theta is in $$the second quadrant, $$\sin \theta \u003e 0$$, so $$\sin \theta = \frac{3}{\sqrt{10}}$$

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

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

Q: A ladder leans against a wall making an angle of $$60^\circ$$ with the ground. If the ladder is 10 meters long, how high up the wall does it reach?

$$h = 10 \times \sin 60^\circ \approx 8.66$$ meters

Question 1

A surveyor measures the angle of elevation from point A to the top of a building as $$35^\circ. If $$point A is 50 meters from the base of the building, what is the height of the building (to the nearest meter)?

Accepted Answer

$$	an 35^\circ = \frac{h}{50}$$, so $$h = 50 	imes 	an 35^\circ \approx 35$$ meters

Question 2

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

Accepted Answer

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

Question 3

Convert $$225^\circ to $$radians.

Accepted Answer

$$225^\circ = \frac{5\pi}{4}$$ radians

Question 4

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

Accepted Answer

$$\omega = \frac{2\pi}{2} = \pi$$ radians/minute

Question 5

Which of the following is a fundamental trigonometric identity?

Accepted Answer

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

Question 6

Given $$\sin 	heta = 0.6$$ and $$	heta is an $$acute angle, what is $$\cos 	heta$$?

Accepted Answer

$$\cos 	heta = \sqrt{1 - (0.6)^2$} = 0.8$$

Question 7

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

Accepted Answer

$$\sin 	heta = \frac{3}{\sqrt{1 + 3^2$}} = \frac{3}{\sqrt{10}}$$, but since $$	heta is in $$the second quadrant, $$\sin 	heta \u003e 0$$, so $$\sin 	heta = \frac{3}{\sqrt{10}}$$

Question 8

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

Accepted Answer

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

Question 9

A ladder leans against a wall making an angle of $$60^\circ$$ with the ground. If the ladder is 10 meters long, how high up the wall does it reach?

Accepted Answer

$$h = 10 	imes \sin 60^\circ \approx 8.66$$ meters

Question 10

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

Accepted Answer

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

Question 11

If $$\cos 	heta = -0.4$$ and $$	heta is in $$the third quadrant, what is $$\sin 	heta$$?

Accepted Answer

$$\sin 	heta = -\sqrt{1 - (-0.4)^2$} = -\sqrt{1 - 0.16} = -\sqrt{0.84} \approx -0.916$$

Question 12

A car wheel has a diameter of 0.6 meters and rotates at 120 revolutions per minute. What is the linear speed of a point on the rim (in meters per minute)?

Accepted Answer

$$v = r \omega = 0.3 	imes (120 	imes 2\pi) = 0.3 	imes 240\pi \approx 226$$ meters/minute

Question 13

Which of the following is the correct Law of Sines for triangle $$ABC$$?

Accepted Answer

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

Question 14

Given $$\sin \alpha = 0.8$$ and $$\cos \beta = 0.6$$, with both angles acute, what is $$\sin(\alpha + \beta)$$?

Accepted Answer

$$\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta = 0.8 	imes 0.6 + 0.6 	imes 0.8 = 0.48 + 0.48 = 0.96$$

Question 15

Which of the following is the correct expression for the period of the function $$y = 3 \sin(2x)$$?

Accepted Answer

$$	ext{Period} = \frac{2\pi}{2} = \pi$$

Question 16

If $$	an 	heta = \frac{5}{12}$$ and $$	heta is an $$acute angle, what is $$\sin 	heta$$?

Accepted Answer

$$\sin 	heta = \frac{5}{\sqrt{5^2$ + 12^2$}} = \frac{5}{13}$$

Question 17

Which of the following is the correct half-angle identity for sine?

Accepted Answer

$$\sin \frac{	heta}{2} = \pm \sqrt{\frac{1 - \cos 	heta}{2}}$$

Question 18

A triangle has sides $$a = 7$$, $$b = 10$$, and $$c = 12. $$What is the measure of angle $$C$$ using the Law of Cosines?

Accepted Answer

$$\cos C = \frac{a^2$ + b^2$ - c^2$}{2ab} = \frac{49 + 100 - 144}{2 	imes 7 	imes 10} = \frac{5}{140} \approx 0.036$$, so $$C \approx 87.9^\circ$$

MAT 103 Trigonometry Syllabus and Course Structure Study Guide

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