Exercises: Law of Sines - ACT Math

A

Recall / Warm-Up

1.

In a right triangle, if the side opposite an angle is 5 and the hypotenuse is 13, what is the sine of that angle?

A.

5/13

B.

12/13

C.

5/12

D.

13/5

2.

If $\sin(\theta) = 0.6$ , what is $\theta$ to the nearest degree?

A.

37°

B.

53°

C.

30°

D.

60°

3.

In triangle ABC, angle A = 50° and angle B = 70°. What is angle C in degrees?

B

Fluency Practice

1.

In triangle ABC, angle A = 40°, angle B = 75°, and side $a = 10$ . Find side $b$ . Round to the nearest hundredth.

2.

In triangle DEF, angle D = 35°, angle F = 80°, and side $f = 20$ . Find side $d$ . Round to the nearest hundredth.

3.

In triangle PQR, angle P = 50°, angle Q = 60°, and side $p = 14$ . What is side $q$ ? Round to the nearest tenth.

A.

15.8

B.

11.3

C.

12.4

D.

18.3

4.

In triangle ABC, angle A = 30°, side $a = 8$ , and side $b = 10$ . Find angle B in degrees. Round to the nearest tenth.

5.

In triangle XYZ, angle X = 45°, angle Y = 100°, and side $y = 25$ . Find side $x$ . Round to the nearest tenth.

C

Varied Practice

D

Word Problems / Application

1.

Two fire lookout towers are 30 miles apart. Tower A spots a fire at an angle of 48° from the line connecting the two towers. Tower B spots the same fire at an angle of 63° from the same line.

How far is the fire from Tower A? Round to the nearest tenth of a mile.

2.

A surveyor stands at point P and measures the angle to two landmarks, Q and R. The angle QPR = 35°. She knows that PR = 200 m and QR = 120 m. She wants to find angle PQR using the Law of Sines.

How many possible triangles can be formed with this information?

A.

No triangle — the given measurements are impossible

B.

Exactly one triangle

C.

Exactly two triangles

D.

Cannot be determined without more information

3.

A boat leaves port and sails 18 km on a course that makes a 52° angle with the shoreline. The boat then turns and heads toward a lighthouse that is down the coast. From the lighthouse, the angle between the shoreline and the line of sight to the port is 44°.

How far is the boat from the lighthouse after it turns? Round to the nearest tenth of a kilometer.

E

Error Analysis

1.

Marcus solved this problem:

"In triangle ABC, angle A = 50°, angle B = 65°, and side b = 18. Find side a."

Marcus's work:

$\frac{a}{\sin B} = \frac{b}{\sin A}$
$a = \frac{18 \times \sin 65\degree}{\sin 50\degree}$
$a = \frac{18 \times 0.9063}{0.7660} = 21.3$

What error did Marcus make, and what is the correct value of side a?

A.

Marcus paired side a with angle B instead of angle A. The correct answer is approximately 15.2.

B.

Marcus forgot to find the third angle first. The correct answer is approximately 19.4.

C.

Marcus's calculator was in radian mode. The correct answer is approximately 21.3.

D.

Marcus's work is correct. Side a = 21.3.

2.

Taylor solved this problem:

"In triangle ABC, angle A = 25°, side a = 6, and side b = 11. Find angle B."

Taylor's work:

$\sin B = \frac{11 \times \sin 25\degree}{6} = \frac{11 \times 0.4226}{6} = 0.7748$
$B = \arcsin(0.7748) = 50.8\degree$
$C = 180\degree - 25\degree - 50.8\degree = 104.2\degree$
"There is one triangle."

What error did Taylor make?

A.

Taylor forgot to check if the supplement of angle B also forms a valid triangle.

B.

Taylor should have used the Law of Cosines because this is SSA.

C.

Taylor's computation of sin B is wrong.

D.

Taylor's answer is correct — there is exactly one triangle.

F

Challenge / Extension

1.

In triangle ABC, angle A = 40°, side $a = 5$ , and side $b = 12$ . How many triangles can be formed?

A.

No triangle

B.

Exactly one triangle

C.

Exactly two triangles

D.

Cannot be determined

2.

In triangle ABC, angle A = 42°, angle B = 73°, and side $c = 20$ . Find the area of the triangle. Round to the nearest tenth.

Hint: The area of a triangle is $\frac{1}{2}ab\sin C$ .