Q: A radioactive substance decays according to the exponential model $$N(t) = N_0$ e$$^{-kt}$$. If the initial amount is N_0$ = 100$$ grams and the decay constant is $$k = 0.05$$, how much remains after 20 years?

$$N(20) = 100 e$$^{-0.05 \times 20}$$ = 100 e$$^{-1}$$ \approx 36.79$$ grams

Q: A Ferris wheel has a radius of 20 meters. What is the length of the arc subtended by a central angle of $$60^\circ$$?

$$\text{Arc length} = 20 \times \frac{60}{360} \times 2\pi = 20\pi/3 \approx 20.94$$ meters

Q: Given $$\cos \theta = \frac{1}{2}$$ and $$\theta is in $$quadrant IV, what is $$\sin \theta$$?

$$\sin \theta = -\frac{\sqrt{3}}{2}$$

Q: Which of the following is a fundamental trigonometric identity?

$$\tan \theta = \frac{\sin \theta}{\cos \theta}$$

Q: A triangle has sides $$a = 7$$, $$b = 8$$, and $$c = 9. $$Using the Law of Cosines, what is the measure of angle $$C$$ opposite side $$c$$?

$$\cos C = \frac{7^2$ + 8^2$ - 9^2$}{2 \times 7 \times 8}$$

Q: What is the area of a triangle with sides $$a = 10$$, $$b = 12$$, and included angle $$C = 60^\circ$$?

$$\text{Area} = \frac{1}{2} ab \sin C = \frac{1}{2} \times 10 \times 12 \times \sin 60^\circ = 60 \times \frac{\sqrt{3}}{2} = 30\sqrt{3}$$

Q: Which of the following is the correct polar form for the complex number $$z = 3 + 4i$$?

$$z = 5 \text{cis} \theta$$, where $$\theta = \arctan\left(\frac{4}{3}\right)$$

Question 1

Which of the following is the correct standard form equation for a circle with center at $$(h, k)$$ and radius $$r$$?

Accepted Answer

$$(x - h)^2$ + (y - k)^2$ = r^2$

Question 2

A radioactive substance decays according to the exponential model $$N(t) = N_0$ e$$^{-kt}$$. If the initial amount is N_0$ = 100$$ grams and the decay constant is $$k = 0.05$$, how much remains after 20 years?

Accepted Answer

$$N(20) = 100 e$$^{-0.05 	imes 20}$$ = 100 e$$^{-1}$$ \approx 36.79$$ grams

Question 3

If an angle in standard position has its terminal side passing through the point $$(3, 4)$$, what is the value of $$\sin 	heta$$?

Accepted Answer

$$\sin 	heta = \frac{4}{5}$$

Question 4

A Ferris wheel has a radius of 20 meters. What is the length of the arc subtended by a central angle of $$60^\circ$$?

Accepted Answer

$$	ext{Arc length} = 20 	imes \frac{60}{360} 	imes 2\pi = 20\pi/3 \approx 20.94$$ meters

Question 5

Which of the following is the correct period for the function $$y = \sin(2x)$$?

Accepted Answer

$$\pi$$

Question 6

Given $$\cos 	heta = \frac{1}{2}$$ and $$	heta is in $$quadrant IV, what is $$\sin 	heta$$?

Accepted Answer

$$\sin 	heta = -\frac{\sqrt{3}}{2}$$

Question 7

Which of the following is a fundamental trigonometric identity?

Accepted Answer

$$	an 	heta = \frac{\sin 	heta}{\cos 	heta}$$

Question 8

If $$\sin x = \frac{3}{5}$$ and $$x is in $$quadrant II, what is $$\cos x$$?

Accepted Answer

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

Question 9

Which formula correctly expresses the Law of Sines?

Accepted Answer

$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$

Question 10

A triangle has sides $$a = 7$$, $$b = 8$$, and $$c = 9. $$Using the Law of Cosines, what is the measure of angle $$C$$ opposite side $$c$$?

Accepted Answer

$$\cos C = \frac{7^2$ + 8^2$ - 9^2$}{2 	imes 7 	imes 8}$$

Question 11

Which of the following is the correct double-angle formula for $$\sin(2x)$$?

Accepted Answer

$$\sin(2x) = 2 \sin x \cos x$$

Question 12

What is the area of a triangle with sides $$a = 10$$, $$b = 12$$, and included angle $$C = 60^\circ$$?

Accepted Answer

$$	ext{Area} = \frac{1}{2} ab \sin C = \frac{1}{2} 	imes 10 	imes 12 	imes \sin 60^\circ = 60 	imes \frac{\sqrt{3}}{2} = 30\sqrt{3}$$

Question 13

Which of the following is the correct formula for the dot product of vectors $$\vec{a} = \langle a_1$, a_2$ \rangle$$ and $$\vec{b} = \langle b_1$, b_2$ \rangle$$?

Accepted Answer

$$\vec{a} \cdot \vec{b} = a_1$ b_1$ + a_2$ b_2$

Question 14

Which of the following is the correct polar form for the complex number $$z = 3 + 4i$$?

Accepted Answer

$$z = 5 	ext{cis} 	heta$$, where $$	heta = \arctan\left(\frac{4}{3}ight)$$

Question 15

Which of the following is the correct sum formula for $$\sin(A + B)$$?

Accepted Answer

$$\sin(A + B) = \sin A \cos B + \cos A \sin B$$

Question 16

Solve the system of equations: $$\begin{cases} 2x + 3y = 7 \ x - y = 1 \end{cases}$$

Accepted Answer

$$x = 2, y = 1$$

Question 17

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

Accepted Answer

$$	ext{Area} = \frac{1}{2} r^2$ 	heta$$

Question 18

Which of the following is the correct expression for the inverse sine function?

Accepted Answer

$$y = \sin$$^{-1}$$ x$$

Question 19

A mass on a spring oscillates according to $$y(t) = 5 \cos(3t). $$What is the amplitude and period of the motion?

Accepted Answer

Amplitude: $$5$$, Period: $$\frac{2\pi}{3}$$

MATH 122: Trigonometry and Precalculus – Syllabus and Study Guide

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