Q: A circle has a radius of $$10$$ meters. What is the length of the arc subtended by a central angle of $$\frac{\pi}{4}$$ radians?

$$\frac{5\pi}{2}$$\text{ meters}$

Q: A bicycle wheel with a radius of $$0.35$$ meters rotates through an angle of $$720^\circ. $$What is the distance traveled by a point on the rim?

$$0.35 \times 4\pi$$\text{ meters}$

Q: A ladder $$20$$ feet long leans against a wall, making an angle of $$60^\circ$$ with the ground. How high up the wall does the ladder reach?

$$20 \times \sin(60^\circ)$$\text{ feet}$

Q: A pilot sees a runway at an angle of depression of $$5^\circ$$ from a height of $$1000$$ meters. How far is the plane from the runway horizontally?

$$\frac{1000}{\tan(5^\circ)}$$\text{ meters}$

Q: A Ferris wheel with a radius of $$15$$ meters rotates through $$120^\circ. $$What is the arc length traveled by a passenger?

$$15 \times \frac{2\pi}{3}$$\text{ meters}$

Q: If $$\csc(\theta) = 2$$ and $$\theta is in$$ $$[0, \pi]$$, what is the exact value of $$\theta$$?

$$\frac{\pi}{6}$$\text{ or }\frac{5\pi}{6}$

Q: A clock’s minute hand is $$12 cm $$long. What is the area swept by the minute hand in $$15$$ minutes?

$$\frac{1}{4} \pi \times 12^2$\text{ cm}^2$

Q: A gear with radius $$8 cm $$rotates $$270^\circ. $$What is the angle in radians and the arc length traveled?

Angle: $$\frac{3\pi}{2}$$ radians; Arc length: $$8 \times \frac{3\pi}{2} cm$$

Question 1

Convert $$135^\circ to $$radians. Leave your answer in terms of $$\pi.$$

Accepted Answer

$$\frac{3\pi}{4}$$

Question 2

Convert $$\frac{5\pi}{6}$$ radians to degrees. Round your answer to the nearest hundredth.

Accepted Answer

$$150.00^\circ$$

Question 3

Find the exact value of $$\sin\left(\frac{\pi}{3}\right).$$

Accepted Answer

$$\frac{\sqrt{3}}{2}$$

Question 4

A circle has a radius of $$10$$ meters. What is the length of the arc subtended by a central angle of $$\frac{\pi}{4}$$ radians?

Accepted Answer

$$\frac{5\pi}{2}$$	ext{ meters}$

Question 5

Two cities lie on the same north-south line and are $$500$$ miles apart. If the Earth’s radius is $$3960$$ miles, what is the central angle (in radians) between the cities?

Accepted Answer

$$\frac{500}{3960}$$

Question 6

A bicycle wheel with a radius of $$0.35$$ meters rotates through an angle of $$720^\circ. $$What is the distance traveled by a point on the rim?

Accepted Answer

$$0.35 	imes 4\pi$$	ext{ meters}$

Question 7

Find the area of a sector of a circle with radius $$6 cm $$and central angle $$60^\circ. $$Leave your answer in terms of $$\pi.$$

Accepted Answer

$$6\pi$$	ext{ cm}^2$

Question 8

Use a calculator to find $$\sec(2)$$, where $$2 is in $$radians. Round your answer to two decimal places.

Accepted Answer

$$-2.40$$

Question 9

If $$\cot(	heta) = 3$$ and $$	heta is in $$the interval $$[0, \frac{\pi}{2})$$, what is $$	heta in $$radians? Round to two decimal places.

Accepted Answer

$$0.32$$

Question 10

Find all exact angles $$s in$$ $$[\frac{\pi}{2}, \pi]$$ such that $$\sin s = \frac{\sqrt{2}}{2}.$$

Accepted Answer

$$s = \frac{3\pi}{4}$$

Question 11

Solve the right triangle with $$a = 7$$, $$b = 24. $$Find the hypotenuse $$c$$ and the angles $$A$$ and $$B (in $$degrees, rounded to two decimal places).

Accepted Answer

$$c = 25, A = 16.26^\circ, B = 73.74^\circ$$

Question 12

A ladder $$20$$ feet long leans against a wall, making an angle of $$60^\circ$$ with the ground. How high up the wall does the ladder reach?

Accepted Answer

$$20 	imes \sin(60^\circ)$$	ext{ feet}$

Question 13

A pilot sees a runway at an angle of depression of $$5^\circ$$ from a height of $$1000$$ meters. How far is the plane from the runway horizontally?

Accepted Answer

$$\frac{1000}{	an(5^\circ)}$$	ext{ meters}$

Question 14

Find the exact value of $$\cos\left(\frac{5\pi}{6}\right).$$

Accepted Answer

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

Question 15

A Ferris wheel with a radius of $$15$$ meters rotates through $$120^\circ. $$What is the arc length traveled by a passenger?

Accepted Answer

$$15 	imes \frac{2\pi}{3}$$	ext{ meters}$

Question 16

If $$\csc(	heta) = 2$$ and $$	heta is in$$ $$[0, \pi]$$, what is the exact value of $$	heta$$?

Accepted Answer

$$\frac{\pi}{6}$$	ext{ or }\frac{5\pi}{6}$

Question 17

A clock’s minute hand is $$12 cm $$long. What is the area swept by the minute hand in $$15$$ minutes?

Accepted Answer

$$\frac{1}{4} \pi 	imes 12^2$	ext{ cm}^2$

Question 18

Find the exact value of $$	an\left(\frac{\pi}{4}ight).$$

Accepted Answer

$$1$$

Question 19

A gear with radius $$8 cm $$rotates $$270^\circ. $$What is the angle in radians and the arc length traveled?

Accepted Answer

Angle: $$\frac{3\pi}{2}$$ radians; Arc length: $$8 	imes \frac{3\pi}{2} cm$$

Trigonometry Test #2 Review – Step-by-Step Study Guidance

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