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 on 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.
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