Exercises: Law of Cosines - ACT Math

A

Recall / Warm-Up

1.

In a right triangle with legs 5 and 12, what is the length of the hypotenuse?

A.

7

B.

13

C.

17

D.

169

2.

What is the value of $\cos(60\degree)$ ?

A.

0

B.

0.5

C.

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

D.

1

3.

The three angles of a triangle always sum to:

A.

$90\degree$

B.

$180\degree$

C.

$270\degree$

D.

$360\degree$

B

Fluency Practice

1.

In triangle $ABC$ , $a = 6$ , $b = 10$ , and $C = 60\degree$ . Find side $c$ . Round to the nearest tenth.

A.

7.2

B.

8.7

C.

76.0

D.

11.7

2.

In triangle $PQR$ , $p = 8$ , $q = 5$ , and $R = 40\degree$ . Find side $r$ . Round to the nearest tenth.

3.

In triangle $ABC$ , $a = 7$ , $b = 9$ , and $C = 120\degree$ . Find side $c$ . Round to the nearest tenth.

A.

13.9

B.

11.4

C.

8.0

D.

193.0

4.

In triangle $ABC$ , $a = 5$ , $b = 8$ , and $c = 9$ . Find angle $C$ . Round to the nearest degree.

5.

In triangle $DEF$ , $d = 11$ , $e = 7$ , and $f = 6$ . Find angle $D$ . Round to the nearest degree.

C

Varied Practice

D

Word Problems / Application

1.

Two hikers start at the same point. One walks 4 km due east, and the other walks 6 km in a direction that makes a $50\degree$ angle with the first hiker's path.

How far apart are the two hikers? Round to the nearest tenth.

2.

A triangular plot of land has sides measuring 120 m, 150 m, and 200 m.

Find the measure of the largest angle of the plot. Round to the nearest degree.

3.

A ship sails 10 km from port $A$ to port $B$ . It then turns $65\degree$ from its original heading and sails 14 km to port $C$ .

1.

Find the distance from port $A$ to port $C$ . Round to the nearest tenth.

2.

Using your answer from part (a), find the angle at port $A$ (the angle $BAC$ ). Round to the nearest degree.

E

Error Analysis

1.

Mia solved this problem:

"In triangle $ABC$ , $a = 8$ , $b = 5$ , and $C = 110\degree$ . Find side $c$ ."

Mia's work:

$c^2 = 8^2 + 5^2 - 2(8)(5)\cos(110\degree)$
$\cos(110\degree) = -0.342$
$c^2 = 64 + 25 - 80(-0.342) = 89 - (-27.36) = 89 - 27.36 = 61.64$
$c = \sqrt{61.64} \approx 7.9$

What error did Mia make, and what is the correct answer?

A.

Mia subtracted instead of adding when the cosine was negative. The correct answer is $c \approx 10.8$ .

B.

Mia used the wrong cosine value. $\cos(110\degree) = 0.342$ , so $c \approx 7.8$ .

C.

Mia forgot to take the square root. The correct answer is 61.64.

D.

Mia's answer of $c \approx 7.9$ is correct.

2.

Jake solved this problem:

"In triangle $ABC$ , $a = 6$ , $b = 9$ , $c = 10$ . Find angle $A$ ."

Jake's work:

$\cos(A) = \frac{a^2 + b^2 - c^2}{2ab}$
$\cos(A) = \frac{36 + 81 - 100}{2(6)(9)} = \frac{17}{108} \approx 0.157$
$A = \arccos(0.157) \approx 81\degree$

What error did Jake make, and what is the correct answer?

A.

Jake used the formula for angle $C$ , not angle $A$ . The correct answer is $A \approx 36\degree$ .

B.

Jake forgot to take the square root. The correct answer is 17.

C.

Jake's answer of $A \approx 81\degree$ is correct.

D.

Jake should have used the Law of Sines instead. The Law of Cosines cannot find angles.

F

Challenge / Extension

1.

In triangle $ABC$ , $a = 7$ , $b = 10$ , and $C = 55\degree$ . Find all three angles of the triangle. What is the measure of angle $A$ ? Round to the nearest degree.

2.

The $-2ab\cos(C)$ term in the Law of Cosines is sometimes called the "correction factor." Explain what this term corrects for, and describe how the sign of $\cos(C)$ affects the length of side $c$ compared to what the Pythagorean theorem predicts.