# 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 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 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.