__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.