Question 1

Given $$\cos 	heta = \frac{8}{17}$$ and $$\frac{3\pi}{2} \u003c 	heta \u003c 2\pi$$, find $$\sin(2	heta).$$

Accepted Answer

$$-\frac{240}{289}$$

Question 2

If $$\csc 	heta = -\frac{8}{3}$$ and $$	an 	heta \u003e 0$$, find $$\cos(2	heta).$$

Accepted Answer

$$-\frac{1}{8}$$

Question 3

Find the exact value of $$3 \sin^2$ \left[ 2 \cos$$^{-1}$$ \left( \frac{4}{5} \right) \right].$$

Accepted Answer

$$\frac{12}{25}$$

Question 4

Use the Half-Angle Formula to find the exact value of $$\sin(165^\circ).$$

Accepted Answer

$$\frac{1}{2}\sqrt{2 + \sqrt{3}}$$

Question 5

Given $$\csc 	heta = -\frac{3}{2}$$ and $$	an 	heta \u003e 0$$, find $$\cos \left( \frac{	heta}{2} ight).$$

Accepted Answer

$$-\frac{\sqrt{18 - \sqrt{5}}}{6}$$

Question 6

Express $$\sin(50^\circ) \sin(40^\circ) as a $$sum containing only sines or cosines.

Accepted Answer

$$\frac{1}{2}[\cos(40^\circ) - \cos(140^\circ)]$$

Question 7

Complete the identity: $$\sin(50^\circ) \sin(40^\circ) = ?$$

Accepted Answer

$$\frac{1}{2}[\cos(10^\circ) - \cos(90^\circ)]$$

Question 8

Express $$\sin \left( 110^\circ \right) + \sin \left( 90^\circ \right) as a $$product of sines and/or cosines.

Accepted Answer

$$2 \sin \left( \frac{110^\circ}{2} \right) \cos \left( \frac{20^\circ}{2} \right)$$

Question 9

A new homeowner has a triangular-shaped backyard. Two of the three sides measure 85 ft and 80 ft and form an included angle of $$125^\circ. $$What is the area of the yard, rounded to the nearest square foot?

Accepted Answer

$$2130 sq. ft$$

Question 10

Find the area of a triangle with $$a = 6$$, $$b = 6$$, $$c = 7. $$Round your answer to two decimal places.

Accepted Answer

$$17.06$$

Question 11

A room in the shape of a triangle has sides of length 5 yd, 10 yd, and 15 yd. If carpeting costs $$5.58/yd^2$$ and padding costs $$2.52/yd^2$$, how much to the nearest dollar will it cost to carpet the room, assuming that there is no waste?

Accepted Answer

$$787$$

Question 12

John (whose line of sight is 6 ft above horizontal) is trying to estimate the height of a tall oak tree. He first measures the angle of elevation from where he is looking as $$25^\circ. He $$walks 30 ft closer to the tree and finds that the angle of elevation has increased by $$12^\circ. $$Estimate the height of the tree rounded to the nearest whole number.

Accepted Answer

$$67 ft$$

Question 13

Solve the triangle: $$a = 8$$, $$b = 7$$, $$\gamma = 52^\circ. $$Find $$c$$, $$\alpha$$, and $$\beta.$$

Accepted Answer

$$c = 10.53$$, $$\alpha = 48.81^\circ$$, $$\beta = 41.19^\circ$$

Question 14

A ship captain travels 250 miles from island A and then finds that he is off course and 150 miles from island B. What angle, in degrees, must he turn through to head straight for island B? Round your answer to two decimal places.

Accepted Answer

$$55.08^\circ$$

Question 15

In flying the 76 miles from Champaign to Pontiac, a student pilot sets a heading that is $$8^\circ$$ off course and maintains an average speed of 80 miles per hour. After 15 minutes, the instructor notes the course error and tells the student to correct his heading. Through what angle will the plane move to correct the heading and how many miles away is Pontiac when the plane turns?

Accepted Answer

$$10.87^\circ$$, $$63.4 mi$$

Question 16

A ship has sailed due west from the port and its bearing from the lighthouse is N37$$^\circE. $$The bearing of the port from the lighthouse is N53$$^\circE. $$How far has the ship sailed from the port? Round your answer to the nearest 0.1 km.

Accepted Answer

$$3.5 km$$

Question 17

Solve the triangle: $$a = 3$$, $$c = 4$$, $$\beta = 80^\circ. $$Find $$b$$, $$\alpha$$, and $$\gamma.$$

Accepted Answer

$$b = 5.56$$, $$\alpha = 59.6^\circ$$, $$\gamma = 40.4^\circ$$

Question 18

Given $$\sin(50^\circ) \sin(40^\circ)$$, which of the following is equivalent?

Accepted Answer

$$\frac{1}{2}[\cos(10^\circ) - \cos(90^\circ)]$$

Question 19

Complete the identity: $$\sin(50^\circ) \sin(80^\circ) = \frac{1}{2}[\cos(30^\circ) - \cos(130^\circ)]. $$What is $$\sin(50^\circ) \sin(80^\circ)$$ equal to?

Accepted Answer

$$\frac{1}{2}[\cos(30^\circ) - \cos(130^\circ)]$$

Precalculus Trigonometry: Identities, Formulas, and Applications Study Guide

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