Q: A bicycle with a tire radius of $$32 cm is $$being ridden at $$180$$ revolutions per minute. How far does the bicycle travel in $$1$$ minute?

$$\text{Distance} = 2\pi r \times \text{revolutions} = 2\pi \times 32 \times 180 = 36,230.4 cm$$

Q: What is the radius $$r of a $$sector with arc length $$s = 5 cm $$and central angle $$\theta = 60^\circ$$?

$$r = \dfrac{s}{\theta}$$, with $$\theta in $$radians: $$\theta = \dfrac{\pi}{3}$$, so $$r = \dfrac{5}{\pi/3} = \dfrac{15}{\pi} cm$

Question 1

Given a right triangle with legs $$a = 7$$ and $$b = 24$$, what is the length of the hypotenuse $$c$$?

Accepted Answer

$$c = \sqrt{7^2$ + 24^2$} = 25$$

Question 2

Find the exact value of $$\sin(30^\circ).$$

Accepted Answer

$$\sin(30^\circ) = \dfrac{1}{2}$$

Question 3

A bicycle with a tire radius of $$32 cm is $$being ridden at $$180$$ revolutions per minute. How far does the bicycle travel in $$1$$ minute?

Accepted Answer

$$	ext{Distance} = 2\pi r 	imes 	ext{revolutions} = 2\pi 	imes 32 	imes 180 = 36,230.4 cm$$

Question 4

Given $$	riangle ABC$$ with $$A = 30^\circ$$, $$B = 41^\circ$$, and $$a = 8$$, use the Law of Sines to find $$b.$$

Accepted Answer

$$\dfrac{a}{\sin A} = \dfrac{b}{\sin B} \implies b = 8 \dfrac{\sin 41^\circ}{\sin 30^\circ} \approx 10.5$$

Question 5

What is the radius $$r of a $$sector with arc length $$s = 5 cm $$and central angle $$	heta = 60^\circ$$?

Accepted Answer

$$r = \dfrac{s}{	heta}$$, with $$	heta in $$radians: $$	heta = \dfrac{\pi}{3}$$, so $$r = \dfrac{5}{\pi/3} = \dfrac{15}{\pi} cm$

Question 6

Convert $$z = 2	ext{cis}\left(\dfrac{\pi}{4}ight) to $$rectangular form.

Accepted Answer

$$z = 2(\cos\dfrac{\pi}{4} + i\sin\dfrac{\pi}{4}) = 2\left(\dfrac{\sqrt{2}}{2} + i\dfrac{\sqrt{2}}{2}\right) = \sqrt{2} + i\sqrt{2}$$

Question 7

If $$y = \sin(x - \pi)$$, which of the following best describes the transformation compared to $$y = \sin x$$?

Accepted Answer

A phase shift of $$\pi$$ units to the right

Question 8

Evaluate $$\cos(135^\circ).$$

Accepted Answer

$$\cos(135^\circ) = -\dfrac{\sqrt{2}}{2}$$

Question 9

Solve for $$x$$: $$2\sin x - \sqrt{2} = 0$$ for $$0 \leq x \u003c 2\pi.$$

Accepted Answer

$$x = \dfrac{\pi}{4}, \dfrac{3\pi}{4}$$

Question 10

Given $$\sec x = 1$$, what are the solutions for $$x in$$ $$0 \leq x \u003c 2\pi$$?

Accepted Answer

$$x = 0, 2\pi$$

Question 11

A triangle has sides $$a = 8$$, $$b = 6$$, and angle $$A = 30^\circ. $$Use the Law of Sines to find angle $$B.$$

Accepted Answer

$$\dfrac{8}{\sin 30^\circ} = \dfrac{6}{\sin B} \implies \sin B = \dfrac{6 \sin 30^\circ}{8} = 0.375 \implies B \approx 22^\circ$$

Question 12

Which of the following is the correct formula for the area $$A of a $$triangle given two sides $$a$$, $$b$$ and the included angle $$C$$?

Accepted Answer

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

Question 13

Convert $$z = 5	ext{cis}\left(\dfrac{\pi}{3}ight) to $$rectangular form.

Accepted Answer

$$z = 5\left(\cos\dfrac{\pi}{3} + i\sin\dfrac{\pi}{3}\right) = 5\left(\dfrac{1}{2} + i\dfrac{\sqrt{3}}{2}\right) = 2.5 + 2.5i\sqrt{3}$$

Question 14

Given $$y = \sec(x)$$, which of the following statements is true about its graph?

Accepted Answer

It has vertical asymptotes at $$x = \dfrac{\pi}{2} + k\pi$$, $$k \in \mathbb{Z}$$

Question 15

If $$	an 	heta = \dfrac{3}{4}$$ and $$	heta is in $$the first quadrant, what is $$\sin 	heta$$?

Accepted Answer

$$\sin 	heta = \dfrac{3}{5}$$

Question 16

A sector of a circle has radius $$r = 10 cm $$and central angle $$	heta = 120^\circ. $$What is the area of the sector?

Accepted Answer

$$A = \dfrac{1}{2} r^2$ 	heta ($$with $$	heta in $$radians): $$	heta = \dfrac{2\pi}{3}$$, so $$A = \dfrac{1}{2} 	imes 100 	imes \dfrac{2\pi}{3} = \dfrac{100\pi}{3} cm$^2$$

Question 17

Solve for $$x in$$ $$\sin x = 0$$ for $$0 \leq x \u003c 2\pi.$$

Accepted Answer

$$x = 0, \pi, 2\pi$$

Question 18

Given $$a = 10$$, $$b = 8$$, and $$C = 45^\circ$$, find the area of the triangle.

Accepted Answer

$$A = \dfrac{1}{2}ab\sin C = \dfrac{1}{2} 	imes 10 	imes 8 	imes \sin 45^\circ = 40 	imes \dfrac{\sqrt{2}}{2} = 20\sqrt{2}$$

Comprehensive Trigonometry Study Notes: Functions, Triangles, Radian Measure, and Applications

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