A set of problems, that may be solved using the trigonometric ratios, is presented. Detailed solutions and explanations are included.

**6 - **The cotangent of angle A = cot (A)
= side adjacent to angle A / side opposite angle A = b / a

## Problems

__Problem 1__

Given the right triangle below, find

sin A, cos A, tan A, sec A, csc A and cot A.

__Solution to Problem 1:__

First we need to find the hypotenuse using Pythagora's theorem.

(hypotenuse)^{ 2} = 8^{ 2} + 6^{ 2} = 100

and hypotenuse = 10

We now use the definitions of the six trigonometric ratios given above to find sin A, cos A, tan A, sec A, csc A and cot A.

sin A = side opposite angle A / hypotenuse = 8 / 10 = 4 / 5

cos (A) = side adjacent to angle A / hypotenuse = 6 / 10 = 3 / 5

tan (A) = side opposite angle A / side adjacent to angle A

= 8 / 6 = 4 / 3

sec (A) = hypotenuse / side adjacent to angle A = 10 / 6

= 5 / 3

csc (A) = hypotenuse / side opposite to angle A

= 10 / 8 = 5 / 4

cot (A) = side adjacent to angle A / side opposite angle A

= 6 / 8 = 3 / 4

__Problem 2__

Find c in the figure below.

__Solution to Problem 2:__

We are given angle A and the side opposite to it with c the hypotenuse. The sine ratio gives a relationship between the angle, the side opposite to it and the hypotenuse as follows

sin A = opposite / hypotenuse

Angle A and opposite side are known, hence

sin 31^{ o} = 5.12 / c

Solve for c

c = 5.12 / sin 31^{ o}

and use a calculator to obtain

c (approximately) = 9.94

__Problem 3__

If x is an acute angle of a right triangle and sin x = 3 / 7, find the exact value of the trigonometric functions cos x and cot x.

__Solution to Problem 3:__

If sin x = opposite / hypotenuse = 3 / 7, then we can say that opposite = 3 and hypotenuse = 7 and find the adjacent side using Pythagora's theorem.

hypotenuse^{ 2} = adjacent^{ 2} + opposite^{ 2}

7^{ 2} = adjacent^{ 2} + 3^{ 2}

adjacent = √ (40) = 2 √ (10)

We now use trigonometric ratios to find

cos x = adjacent / hypotenuse = 2 √ (10) / 7

cot x = adjacent / opposite = 2 √ (10) / 3

__Problem 4__

Find the exact values of x and y.

__Solution to Problem 4:__

The sine function involves x and the hypotenuse as follows.

sin 30^{ o} = x / 10

Use sin 30^{ o} = 1 / 2 ( see