# Law of Cosines Worksheet

Law of Cosines Worksheet
• Page 1
1.
In triangle ABC, if $\angle$A = $\theta$ = ${55}^{°}$, $b$ = 19, and $c$ = 15, then find the length of $a$ to two significant digits. [Simplify using calculator.]

 a. 37.2 b. 16.1 c. 42.2 d. 27.2

#### Solution:

a2 = 192 + 152 Ã¢â‚¬â€œ 2(19)(15) cos 55°
[Use law of cosines:
a2 = b2 + c2 - 2bc cos A.]

a2 259.061680

a 16.1, to two significant digits.
[Simplify using calculator.]

2.
In triangle DEF, if $\angle$D = $\theta$ = 45°, $f$ = 16 and $e$ = 20, then what is the length of $d$ to two significant digits?

 a. 34.76 b. 32.96 c. 19.26 d. 14.26

#### Solution:

d2 = 202 + 162 - 2(20) (16) cos 45°
[Using law of cosines:
d2 = e2 + f2 - 2ef cos D.]

d2 » 203.451520

d = 14.26, to two significant digits
[Simplify using calculator.]

3.
In triangle PQR, if $\angle$R = $\theta$ = 69°, $p$ = 18, and $q$ = 15, then find the length of $r$ to two significant digits.

 a. 19.58 b. 24.35 c. 18.85 d. 31.35

#### Solution:

r2 = 182 + 152 Ã¢â‚¬â€œ 2(18)(15) cos 69°
[Use law of cosines:
r2 = p2 + q2 Ã¢â‚¬â€œ 2pq cos R.]

r2 355.481280

r = 18.85
[Simplify using calculator.]

4.
In triangle ABC, if $a$ = 14, $b$ = 10, and $c$ = 19, then the measure of angle B to the nearest degree is:

 a. 31° b. 48° c. 80° d. 61°

#### Solution:

Cos B = 192+142-1022(19)(14)
[Use law of cosines:
b2 = c2 + a2 - 2ac Cos B.
Cos B = c2+a2-b22ac.]

Cos B 0.859022
[Simplify using calculator.]

B = 31°

5.
In triangle DEF, if $d$ = 10, $e$ = 10, and $f$ = 12, then what is the measure of the angle F to the nearest degree?

 a. 74° b. 77° c. 44° d. 56°

#### Solution:

Cos F = 102+102-1222(10)(10)
[Use law of cosines:
f2 = d2 + e2 - 2de Cos F.
Cos F = d2+e2-f22de.]

Cos F 0.28
[Simplify using calculator.]

F = 74°

6.
An equilateral triangle ABC is inscribed in the circle with center O and radius 12 cm. Find the perimeter of the triangle using law of cosines to two significant digits.

 a. 50.7 cm b. 16.9 cm c. 19.9 cm d. 288 cm

#### Solution:

Each side makes equal angle at O.

BOC = 13 × 360°
[Sum of angle measures at point O is 360°.]

= 120°

BC2 = 122 + 122 - 2(12)(12) cos 120°
[Use law of cosines:
BC2 = OB2 + OC2 - 2 · OB · OC · cos BOC.]

BC2 = 288
[Simplify using calculator.]

BC = 16.9
[Simplify using calculator.]

Perimeter of triangle ABC = 3 × BC = 3 × 16.9 = 50.7 cm

7.
The sides of a triangle are 3, $\sqrt{6}$ and $\sqrt{5}$, then the largest angle in the triangle to the nearest degree is:
 a. 84° b. 148° c. 74° d. 79°

#### Solution:

Let a = 3, b = 6, c = 5.

The side a is longer side, so the opposite angle to it is larger.

Cos A = (6)2+(5)2-32[2(6)(5)]
[Use law of cosines:
Cos A = b2+c2-a22bc.]

A = 79°

8.
A point R is 24 cm from P and 30 cm from Q. If $\angle$ PRQ = $\theta$ = 68°15′, then the distance between P and Q to three significant digits is:

 a. 38.3 cm b. 77.2 cm c. 30.7 cm d. 115.7 cm

#### Solution:

Using law of cosines: PQ2 = PR2 + QR2 - 2 (PR)(RQ) CosPRQ.

PQ2 = 242 + 302 - 2(24)(30) cos 68°15′

PQ2 940.064
[Simplify using calculator.]

PQ = 30.7
[Simplify using calculator.]

The distance between P and Q is 30.7 cm.

9.
In triangle ABC, if $a$ = 20, $b$ = 13, and $c$ = 11, then the measure of angle A to the nearest degree is:

 a. 113° b. 67° c. 31° d. 149°

#### Solution:

Cos A = 132+112-2022(13)(11)
[Use law of cosines:
a2 = b2 + c2 - 2bcCos A.
Cos A = b2+c2-a22bc.]

Cos A = - 0.384615
[Simplify using calculator.]

A = 113°

10.
In triangle ABC, if $a$ = 7, $b$ = 9, and $c$ = 11, then find the measure of the angle to the nearest degree, opposite to the longer side.
 a. 76° b. 115° c. 55° d. 86°

#### Solution:

The length c is longer side and the angle opposite to it is angle C.

Cos C = 72+92-1122(7)(9)
[Use law of cosines:
Cos C = a2+b2-c22ab.]

Cos C = 0.071428

C = 86°