Q: A car weighing 2610 lb is traveling downhill and experiences a grade resistance of $$-180 lb. $$Using $$F = W \sin \theta$$, what is the grade angle $$\theta to $$the nearest hundredth of a degree?

$$\theta = \arcsin\left(\frac{-180}{2610}\right) \approx -3.95^\circ$$

Q: A rope is wound around a drum of radius $$0.327 m $$through an angle of $$132.6^\circ. $$How much rope is wound, in meters? (Use $$s = r\theta$$ and convert degrees to radians.)

$$s = 0.327 \times \left(\frac{132.6 \pi}{180}\right) \approx 0.757 m$$

Q: Which of the following is the correct formula for the area $$A of a $$sector of a circle with radius $$r$$ and central angle $$\theta (in $$radians)?

$$A = \frac{1}{2} r^2$ \theta$$

Q: Which of the following is the correct reciprocal identity for $$\tan \theta$$?

$$\tan \theta = \frac{1}{\cot \theta}$$

Q: Which of the following is the correct Pythagorean identity?

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

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

$$\text{radians} = \text{degrees} \times \frac{\pi}{180}$$

Question 1

A light ray passes from air (Medium 1) into water (Medium 2), as shown in the first image. If the angle of incidence in air is $$\$$theta_1$$ and the angle of refraction in water is $$\$$theta_2$$, which equation correctly relates the speeds $$c_1$$ and $$c_2 of $$light in the two media to the angles?

Accepted Answer

$$\displaystyle \frac{c_1$}{c_2$} = \frac{\sin \$$theta_1$$}{\sin \$$theta_2$$}$$

Question 2

Refer to Figure 1. If the radius of the circle is $$r$$ and the arc length subtended by angle $$	heta is$$ $$s$$, what is the correct formula for $$	heta in $$radians?

Accepted Answer

$$	heta = \frac{s}{r}$$

Question 3

A car weighing 2610 lb is traveling downhill and experiences a grade resistance of $$-180 lb. $$Using $$F = W \sin 	heta$$, what is the grade angle $$	heta to $$the nearest hundredth of a degree?

Accepted Answer

$$	heta = \arcsin\left(\frac{-180}{2610}ight) \approx -3.95^\circ$$

Question 4

If the angle $$	heta in $$Figure 1 is $$1$$ radian and the radius $$r is$$ $$5$$ units, what is the length of the arc $$s$$?

Accepted Answer

$$s = 5$$ units

Question 5

A rope is wound around a drum of radius $$0.327 m $$through an angle of $$132.6^\circ. $$How much rope is wound, in meters? (Use $$s = r	heta$$ and convert degrees to radians.)

Accepted Answer

$$s = 0.327 	imes \left(\frac{132.6 \pi}{180}ight) \approx 0.757 m$$

Question 6

Which of the following is the correct formula for the area $$A of a $$sector of a circle with radius $$r$$ and central angle $$	heta (in $$radians)?

Accepted Answer

$$A = \frac{1}{2} r^2$ 	heta$$

Question 7

A wheel rotates at $$30$$ revolutions per minute. How many degrees does it rotate through in $$2$$ seconds?

Accepted Answer

$$360^\circ$$

Question 8

If $$\sin 	heta = -\frac{12}{13}$$ and $$	heta is in $$Quadrant IV, what is $$\cos 	heta$$?

Accepted Answer

$$\frac{5}{13}$$

Question 9

Which of the following is the correct reciprocal identity for $$	an 	heta$$?

Accepted Answer

$$	an 	heta = \frac{1}{\cot 	heta}$$

Question 10

A sector of a circle has an area of $$5\pi$$ and a radius of $$5. $$What is the measure of the central angle in radians?

Accepted Answer

$$0.2\pi$$

Question 11

A car travels at $$59$$ mph at an angle of $$16^\circ to a $$radar gun. What is the radar reading to the nearest hundredth?

Accepted Answer

$$59 \cos(16^\circ) \approx 56.71$$ mph

Question 12

If $$	an 	heta = 8/15$$ and $$	heta is in $$Quadrant I, what is $$\sin 	heta$$?

Accepted Answer

$$\frac{8}{17}$$

Question 13

Which of the following is the correct Pythagorean identity?

Accepted Answer

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

Question 14

A circle has a radius of $$10$$ units. What is the length of the arc subtended by a central angle of $$\frac{\pi}{3}$$ radians?

Accepted Answer

$$\frac{10\pi}{3}$$ units

Question 15

If $$\cos 	heta = 0.7256$$ and $$	heta is in$$ $$[0, \frac{\pi}{2}]$$, what is $$	heta in $$radians (rounded to four decimal places)?

Accepted Answer

$$0.7589$$

Question 16

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

Accepted Answer

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

Comprehensive Study Notes for College Trigonometry: Angles, Triangles, and Radian Measure

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