Q: A circle has a radius of $$8$$ meters and an arc length of $$20$$ meters. What is the radian measure of the central angle that intercepts this arc?

$$\theta = \dfrac{20}{8} = 2.5$$ radians

Q: Given the point $$(-3, 4) on $$the terminal side of an angle in standard position, what is $$\cos \theta$$?

$$\cos \theta = \dfrac{-3}{5}$$

Q: Convert the rectangular coordinates $$(-3, 7) to $$polar coordinates $$(r, \theta).$$

$$r = \sqrt{58}$$, \theta = \arctan\left(\dfrac{7}{-3}\right)$

Q: Find the exact value of $$\tan 75^\circ$$ using the angle sum identity.

$$\tan 75^\circ = \dfrac{\tan 45^\circ + \tan 30^\circ}{1 - \tan 45^\circ \tan 30^\circ}$$

Question 1

A circle has a radius of $$8$$ meters and an arc length of $$20$$ meters. What is the radian measure of the central angle that intercepts this arc?

Accepted Answer

$$	heta = \dfrac{20}{8} = 2.5$$ radians

Question 2

Find the area of a sector of a circle with radius $$2$$ centimeters and central angle $$\dfrac{\pi}{3}$$ radians.

Accepted Answer

$$\dfrac{2^2$}{2} \cdot \dfrac{\pi}{3} = \dfrac{4}{2} \cdot \dfrac{\pi}{3} = 2 \cdot \dfrac{\pi}{3} = \dfrac{2\pi}{3} cm$^2$$

Question 3

Given the point $$(-3, 4) on $$the terminal side of an angle in standard position, what is $$\cos 	heta$$?

Accepted Answer

$$\cos 	heta = \dfrac{-3}{5}$$

Question 4

An airplane flying at an altitude of $$8$$ miles is on a flight path that passes directly over an observer. If the angle of elevation from the observer to the plane is $$38^\circ$$, what is the distance from the observer to the plane (to two decimal places)?

Accepted Answer

$$\dfrac{8}{\sin 38^\circ} \approx 12.98$$ miles

Question 5

Find the amplitude, period, and phase shift of the function $$y = -2 \sin(2x + 2\pi).$$

Accepted Answer

Amplitude: $$2$$, Period: $$\pi$$, Phase shift: $$-\pi$$

Question 6

Given $$\sin \alpha = \dfrac{3}{5}$$ and $$\cos \beta = -\dfrac{7}{25}$$, both $$\alpha$$ and $$\beta in $$Quadrant II, find $$\sin(\alpha + \beta).$$

Accepted Answer

$$\sin(\alpha + \beta) = -\dfrac{4}{25}$$

Question 7

Use the Law of Sines to solve for $$B$$ and $$C (to $$two decimal places) given $$a = 30^\circ$$, $$b = 5$$, $$A = 30^\circ.$$

Accepted Answer

$$B = 113.13^\circ$$, $$C = 36.87^\circ$$

Question 8

Convert the rectangular coordinates $$(-3, 7) to $$polar coordinates $$(r, 	heta).$$

Accepted Answer

$$r = \sqrt{58}$$, 	heta = \arctan\left(\dfrac{7}{-3}ight)$

Question 9

Given $$y = 7 \sin 40t$$, what is the period and amplitude?

Accepted Answer

Period: $$\dfrac{2\pi}{40}$$, Amplitude: $$7$$

Question 10

Find the exact value of $$	an 75^\circ$$ using the angle sum identity.

Accepted Answer

$$	an 75^\circ = \dfrac{	an 45^\circ + 	an 30^\circ}{1 - 	an 45^\circ 	an 30^\circ}$$

Question 11

A tree casts a shadow of $$128$$ feet when the angle of elevation of the sun is $$33^\circ. $$What is the height of the tree? (Round to one decimal place.)

Accepted Answer

$$128 \cdot 	an 33^\circ \approx 83.2 ft$$

Question 12

Use the double-angle formula to find the exact value of $$\cos 2	heta$$ when $$	heta = \dfrac{5\pi}{12}.$$

Accepted Answer

$$\cos 2	heta = 2\cos^2$ 	heta - 1$$

Question 13

Given $$a = 30^\circ$$, $$b = 7$$, $$c = 5$$, use the Law of Sines to solve the triangle for the value of $$a. $$Round your answer to two decimal places.

Accepted Answer

$$a = 13.13$$

Question 14

Convert the polar equation $$r = 2\sin 	heta to $$rectangular form.

Accepted Answer

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

Question 15

Use Heron's formula to find the area of a triangle with sides $$a = 7$$, $$b = 8$$, $$c = 5. $$Round your answer to two decimal places.

Accepted Answer

$$\sqrt{10 \cdot (10-7) \cdot (10-8) \cdot (10-5)} = \sqrt{10 \cdot 3 \cdot 2 \cdot 5} = \sqrt{300} \approx 17.32$$

Question 16

A tracking station determines the distance between two aircraft, A and B, using the law of cosines. If $$d_1 = 22$ miles, d_2$ = 35$$ miles, and the angle between them is $$47^\circ$$, what is the distance between the planes? (Round to two decimal places.)

Accepted Answer

$$\sqrt{22^2$ + 35^2$ - 2 \cdot 22 \cdot 35 \cdot \cos 47^\circ} \approx 27.13$$ miles

Question 17

Given $$	an x = 2$$, find $$\sin 2x.$$

Accepted Answer

$$\sin 2x = \dfrac{2 \cdot 2}{1 + 2^2$} = \dfrac{4}{5}$$

Question 18

Write the expression $$\arccos(\sin x) as an $$algebraic expression.

Accepted Answer

$$\arccos(\sin x) = \dfrac{\pi}{2} - x$$

Comprehensive Trigonometry Final Exam Review: Key Concepts and Problem Types

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