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).
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.
More References and Links to Sine and Cosine Lawssine law
Sine Law Calculator and Solver.
Geometry Tutorials, Problems and Interactive Applets.
Cosine Law Problems.
Cosine Law Calculator and Solver. 