Q: Given $$\sin \theta = \frac{7}{8}$$ and $$\theta is an $$acute angle, use the Pythagorean identity $$\sin^2$ \theta + \cos^2$ \theta = 1 to $$find $$\cos \theta. $$Express your answer in simplest radical form.

$$\cos \theta = \frac{\sqrt{15}}{8}$$

Q: Which system of equations has $$(-4, 3) as a $$solution?

$$\begin{cases} -2x + y = 11 \\ 4x + y = -13 \end{cases}$$

Q: Let $$x$$ and $$y be $$two numbers. Four times the first number decreased by the second is $$22. $$The first number increased by four times the second is $$31. $$Write a system of equations and solve for $$x$$ and $$y.$$

$$\begin{cases} 4x - y = 22 \\ x + 4y = 31 \end{cases};\ x = 7,\ y = 6$$

Q: Given a right triangle with legs $$a = 6$$, $$b = 8$$, and hypotenuse $$c = 10$$, find $$\sin \theta$$, $$\cos \theta$$, and $$\tan \theta$$ for the angle $$\theta$$ opposite the side of length $$6.$$

$$\sin \theta = \frac{3}{5},\ \cos \theta = \frac{4}{5},\ \tan \theta = \frac{3}{4}$$

Question 1

Given $$\sin 	heta = \frac{7}{8}$$ and $$	heta is an $$acute angle, use the Pythagorean identity $$\sin^2$ 	heta + \cos^2$ 	heta = 1 to $$find $$\cos 	heta. $$Express your answer in simplest radical form.

Accepted Answer

$$\cos 	heta = \frac{\sqrt{15}}{8}$$

Question 2

Determine whether the ordered pair $$(7, -1) is a $$solution to the system: 
$$\begin{cases} 3x + 4y = 17 \ 2x - 3y = 15 \end{cases}$$

Accepted Answer

It is not a solution.

Question 3

A surveyor wants to find the distance across a lake. From point $$A$$, she measures a $$49^\circ$$ angle to point $$B on $$the opposite shore. She then walks $$710$$ yards to point $$C$$ and measures a $$71^\circ$$ angle back to $$B. $$What is the distance $$a$$ across the lake from $$A to$$ $$B$$? (Round to the nearest whole number.)

Accepted Answer

$$817$$ yards

Question 4

For the linear function $$f(x) = mx + b$$, if $$f(-5) = 15$$ and $$f(6) = -7$$, find the values of $$m$$ and $$b.$$

Accepted Answer

$$m = -2,\ b = 5$$

Question 5

Find the length of the arc $$s on a $$circle of radius $$r = 19$$ inches intercepted by a central angle $$	heta = 175^\circ. $$Express the arc length in terms of $$\pi$$, then round your answer to two decimal places.

Accepted Answer

$$s = \frac{665\pi}{36}$$ inches; $$s \approx 58.03$$ inches

Question 6

Find a positive angle less than $$2\pi$$ that is coterminal with $$-\frac{53\pi}{6}.$$

Accepted Answer

$$\frac{7\pi}{6}$$

Question 7

Which system of equations has $$(-4, 3) as a $$solution?

Accepted Answer

$$\begin{cases} -2x + y = 11 \ 4x + y = -13 \end{cases}$$

Question 8

An isosceles triangle has two angles of equal measure $$x$$ and a third angle $$y. If $$the sum of the interior angles is $$180^\circ$$, what are the measures of $$x$$ and $$y$$?

Accepted Answer

$$x = 46^\circ,\ y = 88^\circ$$

Question 9

Find a cofunction with the same value as $$	an \frac{\pi}{4}.$$

Accepted Answer

$$\cot \frac{\pi}{4}$$

Question 10

Let $$x$$ and $$y be $$two numbers. Four times the first number decreased by the second is $$22. $$The first number increased by four times the second is $$31. $$Write a system of equations and solve for $$x$$ and $$y.$$

Accepted Answer

$$\begin{cases} 4x - y = 22 \ x + 4y = 31 \end{cases};\ x = 7,\ y = 6$$

Question 11

Given a right triangle with legs $$a = 6$$, $$b = 8$$, and hypotenuse $$c = 10$$, find $$\sin 	heta$$, $$\cos 	heta$$, and $$	an 	heta$$ for the angle $$	heta$$ opposite the side of length $$6.$$

Accepted Answer

$$\sin 	heta = \frac{3}{5},\ \cos 	heta = \frac{4}{5},\ 	an 	heta = \frac{3}{4}$$

Question 12

Find the radian measure of the central angle of a circle of radius $$r = 90$$ inches that intercepts an arc of length $$s = 90$$ inches.

Accepted Answer

$$1$$ radian

Question 13

A real-world application: A Ferris wheel has a radius of $$15$$ meters. If a central angle of $$\frac{\pi}{3}$$ radians is swept out, what is the length of the arc traveled by a passenger?

Accepted Answer

$$5\pi$$ meters

Question 14

If $$\sin 	heta = \frac{5}{13}$$ and $$	heta is in $$the first quadrant, find $$\cos 	heta.$$

Accepted Answer

$$\frac{12}{13}$$

Question 15

A ladder leans against a wall making a $$60^\circ$$ angle with the ground. If the foot of the ladder is $$5$$ meters from the wall, how long is the ladder?

Accepted Answer

$$10$$ meters

Precalculus Practice Test 4 – Step-by-Step Guidance

Study Guide - Practice Questions

Study Guide - Flashcards