Let there be a right triangle ABC, right angled at B. Here ∠A (∠CAB) is an acute angle, AC is hypotenuse, side BC is opposite to ∠A and side AB is adjacent to ∠A.
The trigonometric ratios of ∠A (∠A = θ) in right angled ΔABC are defined as:
sin θ = BC/AC
cos θ = AB/AC
tan θ = BC/AB
cosec θ = AC/BC
sec θ = AC/AB
cot θ = AB/BC
When two sides of a right-triangle are given, its third side can be found out by using the Pythagoras theorem. Then you can find the trigonometric ratios of the given angle.
Sometimes you know one trigonometric ratio and you have to find the values of other t-ratios. This can be easily done by using the definition of t-ratios and the Pythagoras
tan θ = sin θ / cos θ
cosec θ is the reciprocal of sin θ
cosec θ = 1 / sin θ or cosec θ . sin θ = 1
sec θ is reciprocal of cos θ
sec θ = 1 / cos θ or sec θ . cos θ = 1
cot θ is reciprocal of tan θ
cot θ = cos θ / sin θ
Thus, cosec θ, sec θ and cot θ are reciprocal of sin θ, cos θ and tan θ respectively.