Q: A ladder leans against a wall, forming a $$60^\circ$$ angle with the ground. If the ladder is 10 ft long, how high up the wall does it reach? (Use $$\sin 60^\circ.)$$

$$10 \times \sin 60^\circ = 10 \times \dfrac{\sqrt{3}}{2} = 5\sqrt{3} ft$$

Q: A triangle has sides $$a = 7$$, $$b = 8$$, and angle $$C = 60^\circ. $$What is the area of the triangle?

$$\dfrac{1}{2} \times 7 \times 8 \times \dfrac{\sqrt{3}}{2} = 14\sqrt{3}$$

Q: Which of the following is equivalent to $$\tan(A+B)$$?

$$\dfrac{\tan A + \tan B}{1 - \tan A \tan B}$$

Q: A circle has a radius of 5 meters. What is the length of the arc subtended by a central angle of $$\dfrac{\pi}{3}$$ radians?

$$s = r\theta = 5 \times \dfrac{\pi}{3} = \dfrac{5\pi}{3}$$\ meters

Q: Given the arithmetic sequence with $$a = 3$$ and $$d = 5$$, what is the 10th term?

$$a_{10} = 3 + (10-1)\times 5 = 48$$

Question 1

Which of the following is the correct factorization of $$a^3$$ - $$b^3$$?

Accepted Answer

$$a^3$$ - $$b^3 = (a$$ - $$b)(a^2 + ab$$ + $$b^2$$)$$

Question 2

A ladder leans against a wall, forming a $$60^\circ$$ angle with the ground. If the ladder is 10 ft long, how high up the wall does it reach? (Use $$\sin 60^\circ.)$$

Accepted Answer

$$10 	imes \sin 60^\circ = 10 	imes \dfrac{\sqrt{3}}{2} = 5\sqrt{3} ft$$

Question 3

Given $$\log_2$ x = 5$$, what is the value of $$x$$?

Accepted Answer

$$x = 32$$

Question 4

Which of the following is the correct formula for the sum of the first $$n$$ terms of a geometric series with first term $$a$$ and common ratio $$r ($$where $$r 
eq 1$$)?

Accepted Answer

$$S_n = a \dfrac{1 - r^n}{1 - r}$$

Question 5

A triangle has sides $$a = 7$$, $$b = 8$$, and angle $$C = 60^\circ. $$What is the area of the triangle?

Accepted Answer

$$\dfrac{1}{2} 	imes 7 	imes 8 	imes \dfrac{\sqrt{3}}{2} = 14\sqrt{3}$$

Question 6

If $$f(x) = 2^x$$, which of the following is $$f(x+3) in $$terms of $$f(x)$$?

Accepted Answer

$$f(x+3) = 8f(x)$$

Question 7

Which of the following is equivalent to $$	an(A+B)$$?

Accepted Answer

$$\dfrac{	an A + 	an B}{1 - 	an A 	an B}$$

Question 8

A circle has a radius of 5 meters. What is the length of the arc subtended by a central angle of $$\dfrac{\pi}{3}$$ radians?

Accepted Answer

$$s = r	heta = 5 	imes \dfrac{\pi}{3} = \dfrac{5\pi}{3}$$\ meters

Question 9

If $$\cos 	heta = \dfrac{3}{5}$$ and $$	heta is in $$the first quadrant, what is $$\sin 	heta$$?

Accepted Answer

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

Question 10

Which of the following is the correct power reduction identity for $$\sin^2$ A$$?

Accepted Answer

$$\sin^2$ A = \dfrac{1 - \cos 2A}{2}$$

Question 11

Given the arithmetic sequence with $$a = 3$$ and $$d = 5$$, what is the 10th term?

Accepted Answer

$$a_{10} = 3 + (10-1)	imes 5 = 48$$

Question 12

If $$\log_3$ x = 4$$, what is $$\log_x 3$$?

Accepted Answer

$$\dfrac{1}{4}$$

Question 13

A triangle has sides $$a = 6$$, $$b = 8$$, $$c = 10. $$What is the area using Heron's formula?

Accepted Answer

$$24$$

Question 14

Which of the following is the correct sum-to-product identity for $$\sin A + \sin B$$?

Accepted Answer

$$2 \sin \left(\dfrac{A+B}{2}\right) \cos \left(\dfrac{A-B}{2}\right)$$

Question 15

If $$	an 	heta = 2$$ and $$	heta is in $$the first quadrant, what is $$\sec 	heta$$?

Accepted Answer

$$\sec 	heta = \sqrt{5}$$

Question 16

Which of the following is the correct logarithm property for $$\log_b(xy)$$?

Accepted Answer

$$\log_b(xy) = \log_b x + \log_b y$$

Question 17

A population of bacteria triples every hour. If the initial population is 100, what is the population after 4 hours?

Accepted Answer

$$100 	imes 3^4$ = 100 	imes 81 = 8100$$

Question 18

Given $$\sin 2A = 2 \sin A \cos A$$, what is $$\sin 60^\circ$$ using this identity?

Accepted Answer

$$\sin 60^\circ = 2 \sin 30^\circ \cos 30^\circ = 2 	imes \dfrac{1}{2} 	imes \dfrac{\sqrt{3}}{2} = \dfrac{\sqrt{3}}{2}$$

Question 19

If $$\cos A = \dfrac{4}{5}$$ and $$A is in $$the first quadrant, what is $$	an A$$?

Accepted Answer

$$	an A = \dfrac{3}{4}$$

Precalculus Reference Sheet: Key Formulas and Identities

Study Guide - Practice Questions

Study Guide - Flashcards