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.


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Updated: 2 April 2013

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