Cosecant Formula
Cosecant is one of the six trigonometric ratios which is also denoted as cosec or csc. The cosecant formula is given by the length of the hypotenuse divided by the length of the opposite side in a right triangle. There is an interesting relationship between the trigonometric ratios cosecant and sin which will be seen below. Let us understand the cosecant formula using the solved example.
What is Cosecant Formula?
For a given right triangle ABC as shown below, where A is an acute angle
AB = Hypotenuse
AC = Side adjacent to angle A and between angle A and right angle
BC = Side opposite to angle A
The cosecant formula is given as follows:
cosec A = hypotenuse / opposite side = AB / BC = c / a
We know that,
sin A = opposite side / hypotenuse
Thus, cosec A in terms of sin A is given by,
cosec A = 1 / sin A = 1 / (a / c) = c / a
Thus, we can say that the trigonometric ratios cosec and sin has a reciprocal relationship among them.
Solved Examples using Cosecant Formula

Example 1: If the side lengths of a right angled triangle ABC is 13 units, 12 units and 5 units, find the value of cosec A.
Solution:
:
Using Cosecant Formula,
cosec A = Hypotenuse / Side opposite to angle A
= AC / BC = 13 / 5
Answer: cosec A = 13 / 5

Example 2: Find cosec P if sin P = 3 / 5.
Solution: Using cosecant formula we know that,
cosec P = 1 / sin P
= 1 / (3 / 5)
= 5 / 3
Answer: cosec P = 5 / 3