Q: A ladder leans against a wall making an angle of $$60^\circ$$ with the ground. If the ladder is 10 ft long, how high up the wall does it reach? (Round to two decimal places)

$$10 \times \sin 60^\circ = 8.66 ft$$

Q: Given triangle $$ABC$$ with $$a = 7$$, $$b = 5$$, and $$C = 60^\circ$$, use the Law of Cosines to find $$c.$$

$$c = \sqrt{7^2$ + 5^2$ - 2 \times 7 \times 5 \times \cos 60^\circ} = 6$$

Q: Prove the identity $$\tan x = \frac{\sin x}{\cos x}$$ and state the reason.

By definition, $$\tan x = \frac{\sin x}{\cos x}$$

Q: Convert the point $$(3, 3)$$ from rectangular to polar coordinates (in radians).

$$(r, \theta) = (3\sqrt{2}, \frac{\pi}{4})$$

Question 1

Given that $$\sin 	heta = \frac{3}{5}$$ and $$	heta is in $$Quadrant II, what is the value of $$\cos 	heta$$?

Accepted Answer

$$\cos 	heta = -\frac{4}{5}$$

Question 2

Solve the equation $$2\sin x - 1 = 0$$ for $$x in $$the interval $$[0, 2\pi).$$

Accepted Answer

$$x = \frac{\pi}{6}, \frac{5\pi}{6}$$

Question 3

A ladder leans against a wall making an angle of $$60^\circ$$ with the ground. If the ladder is 10 ft long, how high up the wall does it reach? (Round to two decimal places)

Accepted Answer

$$10 	imes \sin 60^\circ = 8.66 ft$$

Question 4

Given triangle $$ABC$$ with $$a = 7$$, $$b = 5$$, and $$C = 60^\circ$$, use the Law of Cosines to find $$c.$$

Accepted Answer

$$c = \sqrt{7^2$ + 5^2$ - 2 	imes 7 	imes 5 	imes \cos 60^\circ} = 6$$

Question 5

Find the exact value of $$\arccos\left(\frac{1}{2}\right).$$

Accepted Answer

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

Question 6

Simplify $$\frac{1 - \cos^2$ x}{\sin x}$$ using algebraic techniques.

Accepted Answer

$$\sin x$$

Question 7

Prove the identity $$	an x = \frac{\sin x}{\cos x}$$ and state the reason.

Accepted Answer

By definition, $$	an x = \frac{\sin x}{\cos x}$$

Question 8

Given $$\sin 	heta = \frac{3}{5}$$ and $$	heta in $$Quadrant II, find $$	an 	heta.$$

Accepted Answer

$$	an 	heta = -\frac{3}{4}$$

Question 9

Use the sum identity to find $$\sin(75^\circ).$$

Accepted Answer

$$\sin(75^\circ) = \sin(45^\circ + 30^\circ) = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ$$

Question 10

Convert the parametric equations $$x = 2t$$, $$y = 3t + 1 to a $$rectangular equation.

Accepted Answer

$$y = \frac{3}{2}x + 1$$

Question 11

Convert $$z = 4	ext{cis}\left(\frac{\pi}{2}ight) to$$ $$a + bi$$ form.

Accepted Answer

$$z = 0 + 4i$$

Question 12

Convert the point $$(3, 3)$$ from rectangular to polar coordinates (in radians).

Accepted Answer

$$(r, 	heta) = (3\sqrt{2}, \frac{\pi}{4})$$

Question 13

A pilot flies from point A to point B. The bearing from A to B is $$N30^\circ E$$ and the distance is 100 miles. What are the north and east components of the trip?

Accepted Answer

North: $$100 \cos 30^\circ = 86.60$$ miles, East: $$100 \sin 30^\circ = 50$$ miles

Question 14

Find the amplitude, period, and phase shift of $$y = 3\sin\left(2x - \frac{\pi}{4}\right).$$

Accepted Answer

Amplitude: $$3$$, Period: $$\pi$$, Phase shift: $$\frac{\pi}{8}$$ right

Question 15

Find all solutions to $$\cos x = -\frac{1}{2} in$$ $$[0, 2\pi).$$

Accepted Answer

$$x = \frac{2\pi}{3}, \frac{4\pi}{3}$$

Question 16

Find a coterminal angle for $$-135^\circ$$ between $$0^\circ$$ and $$360^\circ.$$

Accepted Answer

$$225^\circ$$

Question 17

Draw a reference triangle for $$210^\circ$$ and label its sides and reference angle.

Accepted Answer

Reference angle: $$30^\circ$$, sides: opposite $$= 1$$, adjacent $$= \sqrt{3}$$, hypotenuse $$= 2$$

Question 18

A car wheel has a radius of 0.3 m and rotates at 120 revolutions per minute. What is the linear speed of a point on the rim? (Use $$v = r\omega$$, $$\omega = 2\pi f)$$

Accepted Answer

$$v = 0.3 	imes 2\pi 	imes 120 = 226 m/$$min

Comprehensive Review Guide: Trigonometry and Analytic Geometry for Precalculus Final Exam

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