Cosine Law Problems

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.