Q: A surveyor is measuring the height of a building. Standing 50 meters from the base, she measures the angle of elevation to the top as $$35^ ext{o}. $$What is the height of the building, to the nearest meter?

$$50 \times \tan(35^\text{o}) \approx 35$$ meters

Q: If $$\cos(\theta) = \dfrac{1}{2}$$ and $$0^\text{o} \u003c \theta \u003c 180^\text{o}$$, what are the possible values for $$\theta$$?

$$60^\text{o}$$ and $$120^\text{o}$$

Q: A Ferris wheel has a radius of 20 meters. What is the height above the ground of a seat at an angle of $$90^\text{o}$$ from the horizontal, if the center is 25 meters above the ground?

$$25 + 20 \times \sin(90^\text{o}) = 45$$ meters

Q: A vector $$\vec{v}$$ has magnitude 10 and direction $$60^\text{o}$$ from the positive $$x-$$axis. What are its $$x$$ and $$y$$ components?

$$10\cos(60^\text{o}) = 5$$, $$10\sin(60^\text{o}) = 5\sqrt{3}$$

Q: A pilot flies 100 km at a bearing of $$30^\text{o}$$, then 100 km at a bearing of $$120^\text{o}. $$What is the straight-line distance from the starting point?

$$\sqrt{100^2$ + 100^2$ - 2 \times 100 \times 100 \times \cos(90^\text{o})} = \sqrt{20000} \approx 141 km$$

Q: If $$\tan(\theta) = 1$$ and $$0^\text{o} \u003c \theta \u003c 360^\text{o}$$, what are the possible values for $$\theta$$?

$$45^\text{o}$$ and $$225^\text{o}$$

Q: A ramp rises 2 meters vertically over a horizontal distance of 8 meters. What is the angle of elevation of the ramp?

$$\arctan\left(\dfrac{2}{8}\right) \approx 14^\text{o}$$

Question 1

A surveyor is measuring the height of a building. Standing 50 meters from the base, she measures the angle of elevation to the top as $$35^	ext{o}. $$What is the height of the building, to the nearest meter?

Accepted Answer

$$50 	imes 	an(35^	ext{o}) \approx 35$$ meters

Question 2

Which of the following is the correct value for $$\sin(45^	ext{o})$$?

Accepted Answer

$$\dfrac{\sqrt{2}}{2}$$

Question 3

Given a right triangle with legs of length 1 and $$\sqrt{3}$$, what is the measure of the smallest angle?

Accepted Answer

$$30^	ext{o}$$

Question 4

A vector has components $$\langle 3, 4 \rangle. $$What is its magnitude?

Accepted Answer

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

Question 5

Which equation represents the Law of Sines?

Accepted Answer

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

Question 6

If $$\cos(	heta) = \dfrac{1}{2}$$ and $$0^	ext{o} \u003c 	heta \u003c 180^	ext{o}$$, what are the possible values for $$	heta$$?

Accepted Answer

$$60^	ext{o}$$ and $$120^	ext{o}$$

Question 7

A ship sails 10 km east, then 10 km north. What is the straight-line distance from its starting point?

Accepted Answer

$$\sqrt{10^2$ + 10^2$} = \sqrt{200} \approx 14.1 km$$

Question 8

Which of the following is an identity for $$\sin(2x)$$?

Accepted Answer

$$2\sin x \cos x$$

Question 9

A Ferris wheel has a radius of 20 meters. What is the height above the ground of a seat at an angle of $$90^	ext{o}$$ from the horizontal, if the center is 25 meters above the ground?

Accepted Answer

$$25 + 20 	imes \sin(90^	ext{o}) = 45$$ meters

Question 10

Which of the following is the correct value for $$	an(30^	ext{o})$$?

Accepted Answer

$$\dfrac{1}{\sqrt{3}}$$

Question 11

If $$\sin(x) = 0.6$$ and $$x is in $$quadrant II, what is $$\cos(x)$$?

Accepted Answer

$$-\sqrt{1 - 0.36} = -0.8$$

Question 12

Which of the following is the correct formula for the area of a triangle using trigonometry?

Accepted Answer

$$\dfrac{1}{2}ab\sin C$$

Question 13

A vector $$\vec{v}$$ has magnitude 10 and direction $$60^	ext{o}$$ from the positive $$x-$$axis. What are its $$x$$ and $$y$$ components?

Accepted Answer

$$10\cos(60^	ext{o}) = 5$$, $$10\sin(60^	ext{o}) = 5\sqrt{3}$$

Question 14

Which of the following is the correct value for $$\cos(30^	ext{o})$$?

Accepted Answer

$$\dfrac{\sqrt{3}}{2}$$

Question 15

A pilot flies 100 km at a bearing of $$30^	ext{o}$$, then 100 km at a bearing of $$120^	ext{o}. $$What is the straight-line distance from the starting point?

Accepted Answer

$$\sqrt{100^2$ + 100^2$ - 2 	imes 100 	imes 100 	imes \cos(90^	ext{o})} = \sqrt{20000} \approx 141 km$$

Question 16

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

Accepted Answer

$$\cos^2$ x - \sin^2$ x$$

Question 17

If $$	an(	heta) = 1$$ and $$0^	ext{o} \u003c 	heta \u003c 360^	ext{o}$$, what are the possible values for $$	heta$$?

Accepted Answer

$$45^	ext{o}$$ and $$225^	ext{o}$$

Question 18

Which of the following is the correct value for $$\sin(60^	ext{o})$$?

Accepted Answer

$$\dfrac{\sqrt{3}}{2}$$

Question 19

A ramp rises 2 meters vertically over a horizontal distance of 8 meters. What is the angle of elevation of the ramp?

Accepted Answer

$$\arctan\left(\dfrac{2}{8}ight) \approx 14^	ext{o}$$

Precalculus Trigonometry: Syllabus, Key Concepts, and Study Guide

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