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

## Problem 1A 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 2An 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
## Exercises1. 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 Exercises1. A = 129.8 ^{ o} , B = 34.0 ^{ o}, C = 16.2 ^{ o}
2. 89 miles. ## More References to Cosine and Sine Laws and GeometryCosine Law Calculator and SolverGeometry Tutorials, Problems and Interactive Applets. Sine Law Problems. Sine Law Calculator and Solver. |