# Cosecant Formula

## Cosecant Formula

The trigonometric sine function has a reciprocal known as the Cosecant Formula function. One of the primary six trigonometric functions, Cosecant Formula is denoted by the abbreviation csc x or cosec x, where x is the angle. Cosecant in a right-angled triangle is determined by the relationship between the hypotenuse and perpendicular. It is expressed as csc x = 1 / sin x since it is the reciprocal of sine.

Students will examine the idea of the Cosecant Formula function and comprehend its formula in this post. Using its domain and range, students will plot the cosecant graph and investigate the trigonometric identities, values, and characteristics of cosec x. To better understand the uses of the csc x notion, students shall solve a few instances based on it.

### The formula for Cosec x

The Cosecant Formula function has the formula Cosec x = 1 / sin x because it is the reciprocal of the sine function. Additionally, since csc x is the reciprocal of sin x and sin x’s formula is stated as Sin x = Perpendicular / Hypotenuse, students may express the cosecant function’s formula as follows.

Hypotenuse / Perpendicular = Cosec x

What is the Cosecant Formula in Trigonometric Functions?

One of the crucial six trigonometric functions is the cosecant function. It is the reciprocal of the sine function, making it equal to the right-angled triangle’s hypotenuse to perpendicular ratio.

Use the Cosecant Formula or the csc function to determine how to find csc:

Cosec x = Hypotenuse / Perpendicular

The side of a right-angled triangle that is perpendicular to the desired angle θ

is referred to as an angle’s opposing side. Likewise, the side next to the desired angle θ is referred to as the side next to an angle, whereas the hypotenuse side is the side that faces 90 degrees.

Therefore, to determine an angle’s cosecant, identify the side that is closest to the angle first. Once students determined the hypotenuse side, they could divide using the Cosecant Formula:

Cosec x = Hypotenuse / Perpendicular

What is a Cosecant Function in Maths?

Sine has a reciprocal known as Cosecant Formula. There are six crucial trigonometric operations:

• Sine
• Cosine
• Tangent
• Cotangent
• Secant
• Cosecant

It is known as the ratio of a right-angled triangle’s hypotenuse and perpendicular lengths because it is the reciprocal of sin x. Students could take into account a unit circle with the points O in the middle, P along the edge, and Q inside the circle, and unite them as previously demonstrated. The length of OP is 1 unit because the circle is a unit circle. Students can take into account an angle POQ of x degrees. Using the Cosecant Formula definition, they would have the following.

csc x = OP/PQ

= 1/PQ

Students know that the sine function’s reciprocal, the cosecant function, has a formula. Now that students have a better understanding of the Cosecant Formula function, they could look at some of its key characteristics.

• Cosec x’s graph is symmetrical about the x-axis.
• The cosecant function is a peculiar one, with the formula csc (-x) = -csc x.
• The cosecant graph does not have any x-intercepts, meaning that it never crosses the x-axis.
• When sin x is positive, the value of csc x is positive; conversely, when sin x is negative, it is negative.
• Csc X’s period is 2 radians (360 degrees).
• When multiplied by integral multiples of, cosec x is not defined.

### The Sine Function

The sine function in trigonometry is the ratio of the hypotenuse’s length to the opposite side’s length in a right-angled triangle. To determine a right triangle’s unknown angle or sides, students could utilise the sine function.

The sine function for any right triangle with an angle, for instance of Triangle ABC, will be:

Sin α= Opposite/ Hypotenuse

### Examples of Cosecant x Formula

The six trigonometric functions are known to students; the basic functions are sin, cos, and tan, while the secondary functions are sec, cosec, and cot.

The cosecant, Cosec x = 1/ Sin X, is the reciprocal of sin.

Cosecant x Formula examples

Example 1: If Sin x = 4/7, find Cosec X.

The answer is Cosec X = 1/Sin X.

=1/4/7

=7/4

Thus, Cosec X = 7/4.

Students could visit Extramarks to learn all Six Trigonometric Formulas and Functions.