Sine Law to Solve Triangle Problems

Solve triangle problems using the sine law . A tutorial with problems, detailed solutions and exercises with answers. The ambiguous case of the sine law, where two sides and one angle are given, is also considered (problems 3 and 4).

Problem 1

A triangle ABC has angle A = 106 ^o, angle B = 31 ^o and side a = 10 cm. Solve the triangle ABC by finding angle C and sides b and c.(round answers to 1 decimal place).

Solution to Problem 1:

Use the fact that the sum of all three angles of a triangle is equal to 180 ^o to write an equation in C.
A + B + C = 180 ^o
Solve for C.
C = 180 ^o - (A + B) = 43 ^o
Use sine law to write an equation in b.
a / sin(A) = b / sin(B)
Solve for b.
b = a sin (B) / sin(A) = (approximately) 5.4 cm
Use the sine law to write an equation in c.
a / sin(A) = c / sin(C)
Solve for c.
c = a sin (C) / sin(A) = (approximately) 7.1 cm

Problem 2

The angle of elevation to the top C of a building from two points A and B on level ground are 50 degrees and 60 degrees respectively. The distance between points A and B is 30 meters. Points A, B and C are in the same vertical plane. Find the height h of the building(round your answer to the nearest unit).
diagram problem 2

Solution to Problem 2:

We consider triangle ABC. Angle B internal to triangle ABC is equal to
B = 180^o - 60^o = 120^o
In the same triangle, angle C is given by.
C = 180^o - (50^o + 120^o) = 10 ^o
Use sine law to find d.
d / sin(50) = 30 / sin(10)
Solve for d.
d = 30 *sin(50) / sin(10)
We now consider the right triangle.
sin (60) = h / d
Solve for h.
h = d * sin(60)
Substitute d by the expression found above.
h = 30 *sin(50) * sin(60) / sin(10)
Use calculator to approximate h.
h = (approximately) 115 meters.

Problem 3

A triangle ABC has side a = 12 cm, side b = 19 cm and angle A = 80 ^o (angle A is opposite side a). Find side c and angles B and C if possible.(round answers to 1 decimal place).

Solution to Problem 3:

Use sine law to write an equation in sin(B).
a / sin(A) = b / sin(B)
Solve for sin(B).
sin (B) = (b / a) sin(A) = (19/12) sin(80) = (approximately) 1.6
No real angle B satisfies the equation
sin (B) = 1.6
The given problem has no solution.

Problem 4

A triangle ABC has side a = 14 cm, side b = 19 cm and angle A = 32 ^o (angle A is opposite side a). Find side c and angles B and C if possible.(round answers to 1 decimal place).

Solution to Problem 4

Use sine law to write an equation in sin(B).
a / sin(A) = b / sin(B)
Solve for sin(B).
sin (B) = (b / a) sin(A) = (19/14) sin(32) = (approximately) 0.7192
Two angles satisfy the equation sin (B) = 0.7192 and the given problem has two solutions
B1 = 46.0 ^o and B2 = 134 ^o
Solution 1: Find angle C1 corresponding to B1
C1 = 180 - B1 - A = 102 ^o
Solution 1: Find side c1 corresponding to C1
c1 / sin(C1) = a / sin(A)
c1 = 14 sin(102) / sin(32) = (approximately) 25.8 cm
Solution 2: Find angle C2 corresponding to B2
C2 = 180 - B2 - A = 14 ^o
Solution 2: Find side c2 corresponding to C2
c2 / sin(C2) = a / sin(A)
c1 = 14 sin(14) / sin(32) = (approximately) 6.4 cm

Exercises

1. A triangle ABC has angle A = 104 ^o, angle C = 33 ^o and side c = 9 m. Solve the triangle ABC by finding angle B and sides a and b.(round answers to 1 decimal place).
2. Redo problem 2 with the distance between points A and B equal to 50 meters.

Solutions to Above Exercises

1. B = 43 ^o, a = 16.0 m , b = 11.3 m
2. 191 meters.