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

$$50 \times \tan 30^\circ = 50 \times \dfrac{1}{\sqrt{3}} \approx 28.87$$ meters

Q: A Ferris wheel has a radius of 20 meters. If a passenger is at a point making a $$120^\circ$$ angle with the horizontal diameter, what is the vertical height above the center?

$$20 \times \sin 120^\circ = 20 \times \dfrac{\sqrt{3}}{2} \approx 17.32$$ meters

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

$$\tan(\alpha + \beta) = \dfrac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}$$

Q: If $$\vec{u} = \langle 3, 4 \rangle$$ and $$\vec{v} = \langle 1, -2 \rangle$$, what is the dot product $$\vec{u} \cdot \vec{v}$$?

$$3 \times 1 + 4 \times (-2) = 3 - 8 = -5$$

Q: Which of the following is the double angle formula for $$\sin 2\theta$$?

$$\sin 2\theta = 2 \sin \theta \cos \theta$$

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$$

Question 1

Which of the following is the correct value for $$\sin 45^\circ$$?

Accepted Answer

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

Question 2

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

Accepted Answer

$$50 	imes 	an 30^\circ = 50 	imes \dfrac{1}{\sqrt{3}} \approx 28.87$$ meters

Question 3

Which equation represents the unit circle in the $$xy-$$plane?

Accepted Answer

$$x^2$$ + $$y^2 = 1$

Question 4

Given $$\cos 	heta = \dfrac{1}{2}$$ and $$	heta is in $$quadrant IV, what is $$\sin 	heta$$?

Accepted Answer

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

Question 5

A Ferris wheel has a radius of 20 meters. If a passenger is at a point making a $$120^\circ$$ angle with the horizontal diameter, what is the vertical height above the center?

Accepted Answer

$$20 	imes \sin 120^\circ = 20 	imes \dfrac{\sqrt{3}}{2} \approx 17.32$$ meters

Question 6

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

Accepted Answer

$$	an(\alpha + \beta) = \dfrac{	an \alpha + 	an \beta}{1 - 	an \alpha 	an \beta}$$

Question 7

If $$\vec{u} = \langle 3, 4 \rangle$$ and $$\vec{v} = \langle 1, -2 \rangle$$, what is the dot product $$\vec{u} \cdot \vec{v}$$?

Accepted Answer

$$3 	imes 1 + 4 	imes (-2) = 3 - 8 = -5$$

Question 8

Which of the following is the Law of Sines?

Accepted Answer

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

Question 9

A vector $$\vec{w}$$ has magnitude 10 and makes a $$60^\circ$$ angle with the $$x-$$axis. What are its $$x$$ and $$y$$ components?

Accepted Answer

$$x = 10 \cos 60^\circ = 5$$, $$y = 10 \sin 60^\circ = 5\sqrt{3}$$

Question 10

Which of the following is the correct value for $$\cos 30^\circ$$?

Accepted Answer

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

Question 11

A ship sails 100 km east, then 100 km north. What is the straight-line distance from its starting point?

Accepted Answer

$$\sqrt{100^2$ + 100^2$} = \sqrt{20000} = 100\sqrt{2} km$$

Question 12

Which of the following is the double angle formula for $$\sin 2	heta$$?

Accepted Answer

$$\sin 2	heta = 2 \sin 	heta \cos 	heta$$

Question 13

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 14

Which of the following is the correct value for $$	an 60^\circ$$?

Accepted Answer

$$\sqrt{3}$$

Question 15

If $$\sin 	heta = \dfrac{1}{2}$$ and $$	heta is in $$quadrant II, what is $$\cos 	heta$$?

Accepted Answer

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

Question 16

A sound wave can be modeled by $$y = 3 \sin(2\pi t)$$, where $$t is $$time in seconds. What is the amplitude and period of the wave?

Accepted Answer

Amplitude: $$3$$, Period: $$1$$ second

Question 17

Which of the following is the Pythagorean identity?

Accepted Answer

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

Question 18

A plane flies 300 km at a bearing of $$30^\circ$$ north of east. What are its eastward and northward components?

Accepted Answer

East: $$300 \cos 30^\circ = 300 	imes \dfrac{\sqrt{3}}{2} \approx 259.8 km$$, North: $$300 \sin 30^\circ = 150 km$$

Question 19

Which of the following is the correct value for $$\sin 30^\circ$$?

Accepted Answer

$$\dfrac{1}{2}$$

Precalculus Trigonometry: Syllabus, Key Concepts, and Reference Figures

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