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.