Law of Cosines Worksheet

Law of Cosines Worksheet
  • Page 1
 1.  
In triangle ABC, if A = θ = 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 – 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.]


Correct answer : (2)
 2.  
In triangle DEF, if D = θ = 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.]


Correct answer : (4)
 3.  
In triangle PQR, if R = θ = 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 – 2(18)(15) cos 69°
[Use law of cosines:
r2 = p2 + q2 – 2pq cos R.]

r2 355.481280

r = 18.85
[Simplify using calculator.]


Correct answer : (3)
 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°


Correct answer : (1)
 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°


Correct answer : (-1)
 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


Correct answer : (1)
 7.  
The sides of a triangle are 3, 6 and 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°


Correct answer : (4)
 8.  
A point R is 24 cm from P and 30 cm from Q. If PRQ = θ = 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.


Correct answer : (3)
 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°


Correct answer : (1)
 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°


Correct answer : (4)

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