Q: Given $$\tan \theta = \frac{3}{4}$$ and $$\theta is in $$the first quadrant, what is $$\sin 2\theta$$?

$$\sin 2\theta = \frac{24}{25}$$

Q: Which of the following is the correct power-reducing formula for $$\tan^2$ \theta$$?

$$\tan^2$ \theta = \frac{1 - \cos 2\theta}{1 + \cos 2\theta}$$

Q: A population of bacteria triples every 4 hours. If the initial population is 500, what is the population after 12 hours? Use $$P(t) = P_0$ e$$^{kt}$$.

$$P(12) = 500 \times 3$$^{3}$$ = 13,500$$

Question 1

A radioactive substance decays according to the formula $$A(t) = A_0$ e$$^{kt}$$. If a sample initially has 100 grams and decays to 80 grams in 5 years, what is the decay constant k$? (Round your answer to three decimal places.)

Accepted Answer

$$k = -0.045$$

Question 2

A bank offers an annual interest rate of 6% compounded quarterly. If you invest $$\2000$, how much will you have after 10 years? Use the compound interest formula A = P$$ \left(1 + \frac{r}{n}\$$right)^{nt}$.

Accepted Answer

$$A = 2000 \left(1 + \frac{0.06}{4}\$$right)^{40}$$ \approx \3612.22$

Question 3

Which of the following is the correct expression for $$\sin(75^\circ)$$ using the sum formula?

Accepted Answer

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

Question 4

Given $$	an 	heta = \frac{3}{4}$$ and $$	heta is in $$the first quadrant, what is $$\sin 2	heta$$?

Accepted Answer

$$\sin 2	heta = \frac{24}{25}$$

Question 5

A triangle has sides $$a = 7$$, $$b = 10$$, and angle $$C = 120^\circ. $$What is the length of side $$c$$ using the Law of Cosines?

Accepted Answer

$$c = \sqrt{7^2$ + 10^2$ - 2 \cdot 7 \cdot 10 \cos 120^\circ} \approx 15.6$$

Question 6

Which of the following is the correct power-reducing formula for $$	an^2$ 	heta$$?

Accepted Answer

$$	an^2$ 	heta = \frac{1 - \cos 2	heta}{1 + \cos 2	heta}$$

Question 7

If $$\sin \alpha = \frac{5}{13}$$ and $$\alpha is in $$the second quadrant, what is $$\cos \frac{\alpha}{2}$$?

Accepted Answer

$$\cos \frac{\alpha}{2} = \frac{2}{\sqrt{13}}$$

Question 8

A population of bacteria triples every 4 hours. If the initial population is 500, what is the population after 12 hours? Use $$P(t) = P_0$ e$$^{kt}$$.

Accepted Answer

$$P(12) = 500 	imes 3$$^{3}$$ = 13,500$$

Question 9

Which of the following is the correct product-to-sum formula for $$\sin \alpha \cos \beta$$?

Accepted Answer

$$\sin \alpha \cos \beta = \frac{1}{2} [\sin(\alpha + \beta) + \sin(\alpha - \beta)]$$

Question 10

A triangle has angles $$A = 40^\circ$$, $$B = 60^\circ$$, and side $$a = 8. $$Using the Law of Sines, what is the length of side $$b$$?

Accepted Answer

$$b = \frac{8 \sin 60^\circ}{\sin 40^\circ} \approx 10.4$$

Question 11

Which of the following is the correct sum-to-product formula for $$\sin \alpha + \sin \beta$$?

Accepted Answer

$$\sin \alpha + \sin \beta = 2 \sin \left(\frac{\alpha + \beta}{2}\right) \cos \left(\frac{\alpha - \beta}{2}\right)$$

Question 12

If $$	an \alpha = 2$$ and $$	an \beta = 3$$, what is $$	an(\alpha + \beta)$$?

Accepted Answer

$$	an(\alpha + \beta) = \frac{2 + 3}{1 - 2 \cdot 3} = -1$$

Question 13

A medication is administered and its concentration in the bloodstream decreases by 30% every hour. If the initial concentration is 100 mg/L, what is the concentration after 3 hours? (Use $$A(t) = A_0$ e$$^{kt}$$ and find k$ first.)

Accepted Answer

$$A(3) = 100 	imes (0.7)^3$ = 34.3 mg/L$$

Question 14

Which of the following is the correct half-angle formula for $$\sin \frac{\alpha}{2}$$?

Accepted Answer

$$\sin \frac{\alpha}{2} = \pm \sqrt{\frac{1 - \cos \alpha}{2}}$$

Question 15

If $$\cos 	heta = -\frac{3}{5}$$ and $$	heta is in $$the third quadrant, what is $$\sin 	heta$$?

Accepted Answer

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

Question 16

A loan of $$\5000$ is compounded continuously at an annual rate of 4%. What is the amount owed after 7 years? Use A$$ = $$Pe^{rt}$.

Accepted Answer

$$A = 5000 e$$^{0.04 	imes 7}$$ \approx \6663.91$

Question 17

Which of the following is the correct Law of Sines?

Accepted Answer

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

Question 18

If $$\sin \alpha = \frac{3}{5}$$ and $$\alpha is in $$the first quadrant, what is $$\sin(2\alpha)$$?

Accepted Answer

$$\sin(2\alpha) = 2 \cdot \frac{3}{5} \cdot \frac{4}{5} = \frac{24}{25}$$

Precalculus Study Guide: Exponential Functions, Trigonometric Identities, and Laws

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