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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 sine 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
Cosecant is the reciprocal of sine. 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 rightangled 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
= AB / 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

Example 3: Calculate the value of cosec x if angle x is equal to π/4 radians.
Solution: Using trigonometry table,
cosec x = cosec (π/4)
= 1 / sin (π/4)
= 1/(1/√2) = √2
Answer: cosec x = √2
FAQs on Cosecant Formula
What is Cosecant Formula?
Cosecant formula gives the formula for the trigonometric function cosecant for a rightangled triangle. The cosecant formula is given by the ratio of the hypotenuse and perpendicular (opposite side) of a rightangled triangle.
What is the Cosecant Formula in Terms of Sine?
We know that the cosecant function is the reciprocal of the sine function. So, the cosecant formula in terms of sine is given by, cosec x = 1/sin x.
Is Cosecant the Inverse of Sine?
No, cosecant is not the inverse of sine. It is the reciprocal of the sine function. The inverse of sine is sin inverse x or arcsin x.
What is the Cosecant Formula in Terms of Cotangent?
Using trigonometry identity, we have 1 + cot^{2}x = csc^{2}x. So, cosecant formula in terms of cotangent is csc x = ± √(1 + cot^{2}x).
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