Cosine Law Problems

Solve problems using the cosine law; a tutorial with detailed solutions and exercises with answers.

Problem 1

A triangle has sides equal to 5 cm, 10 cm and 7 cm. Find its angles (round answers to 1 decimal place).

Solution to Problem 1:

  • Let us use the figure below and set
    Triangle Problem 1

    a = 10 cm , b = 7 cm and c = 5 cm.
  • We now use cosine law to find the largest angle A.
    a 2 = b 2 + c 2 - 2 b c cos(A)
  • Substitute a, b and c by their values and solve for cos (A)
    cos (A) = [ b 2 + c 2 - a 2 ] / 2 b c
    cos (A) = [ 7 2 + 5 2 - 10 2 ] / (2*7*5)
    cos (A) = [ 7 2 + 5 2 - 10 2 ] / (2*7*5)
    = -13 / 35
  • Use calculator to find angle A and round to 1 decimal place.
    A = arccos(-13 / 35) (approximately) = 111.8 o
  • We may again use the cosine law to find angle B or the sine law. We use the sine law.
    a / sin (A) = b / sin(B)
  • sin(B) is given by.
    sin (B) = (b / a) sin(A) = (7 / 10) sin (111.8 o)
  • Use calculator to find B and round to 1 decimal place.
    B (approximately) = 40.5 o
  • Use the fact that the sum of all angles in a triangle is equal to 180 o to find angle C.
    C (approximately) = 180 o - (40.5 o + 111.8 o)
    = 27.7 o

Problem 2

An aircraft tracking station determines the distance from a common point O to each aircraft and the angle between the aicrafts. If angle O between the two aircrafts is equal to 49 o and the distances from point O to the two aircrafts are 50 km and 72 km, find distance d between the two aircrafts.(round answers to 1 decimal place).

Solution to Problem 2:

  • A diagram to the above problem is shown below
    diagram problem 2

  • The cosine law may be used as follows
    d 2 = 72 2 + 50 2 - 2 (72)(50) cos(49 o)
  • Solve for d and use calculator.
    d = SQRT [72 2 + 50 2 - 2 (72)(50) cos(49 o)]
    (approximately) = 54.4 km

Exercises


1. A triangle has sides equal to 4 m, 11 m and 8 m. Find its angles (round answers to 1 decimal place).
2. A ship leaves port at 1 pm traveling north at the speed of 30 miles/hour. At 3 pm, the ship adjusts its course 20 degrees eastward. How far is the ship from the port at 4pm? (round to the nearest unit).

Solutions to Above Exercises


1. A = 129.8
o , B = 34.0 o, C = 16.2 o
2. 89 miles.

More References to Cosine and Sine Laws and Geometry

Cosine Law Calculator and Solver
Geometry Tutorials, Problems and Interactive Applets.
Sine Law Problems.
Sine Law Calculator and Solver.