The cosecant function ‘or’ **Cosec Theta** is one of the trigonometric functions apart from sine, cosine, tangent, secant, and cotangent. In right-angled trigonometry, the cosecant function is defined as the ratio of the hypotenuse and opposite side.

The mathematical denotation of the sine function is,

\(coses(\theta) = \frac{\text{Hypotenuse Side}}{\text{Opposite Side}}\)Index

**More About Cosec Theta**

The derivative of \(cosec(\theta)\) in calculus is \(-cot(\theta)cosec(\theta)\), and the integral of it is \(\ln|cosec(\theta) – cot(\theta)|\). The reciprocal of \(cosec(\theta)\) is \(sin(\theta)\).

Below is a table of cosecant theta values for different degrees and radians.

Radians | Degree | Cosecant Value |

0 | 0° | \(\infty\) |

\(\frac{\pi}{6}\) | 30° | 2 |

\(\frac{\pi}{4}\) | 45° | \(\sqrt{2}\) |

\(\frac{\pi}{3}\) | 60° | \(\frac{2}{\sqrt{3}}\) |

\(\frac{\pi}{2}\) | 90° | 1 |

\(\pi\) | 180° | \(\infty\) |

\(\frac{3\pi}{2}\) | 270° | -1 |

\(2\pi\) | 360° | \(\infty\) |

**Important Cosec Theta Formula**

Some important properties of the cosecant function and cosec theta formula are:

- \(cosec(-x) = -cosec(x)\)

- \(cosec(90°-x) = sec(x)\)

- \(cosec(x + 2 \pi) = cosec(x)\)

- \(cosec(\pi – x) = cosec(x)\)

- \(cosec^2(x) = 1+ cot^2(x)\)

**Solved Examples**

**Question 1.** **If \(cos(x) = \frac{3}{5}\), calculate the value of \(cosec(x)\).**

**Solution.** Using trigonometric identity,

\(sin^2(x) = 1 – cos^2(x) = 1 – \frac{9}{25} \frac{16}{25}\)

\(sin(x) = \frac{4}{5}\)

As we know, \(cosec(x) = \frac{1}{sin(x)}\)

\(∴ cosec(x) = \frac{5}{4}\)

**Question 2.** **If \(cot(x) = \frac{4}{5}\), calculate the value of \(cosec(x)\).**

**Solution.** Using trigonometric identity,

\(cosec^2(x) = 1 + cot^2(x) = (\frac{4}{5})^2 + 1 = \frac{41}{25}\)

\(∴ cosec(x)= \frac{\sqrt{41}}{5}\)

**FAQs**

**Explain how cosec(-x) = -cosec(x).**As we know, the angle (-x) lies in the 4th quadrant of a graph, and the cosecant is negative in this quadrant. Hence, this shows that cosec(-x) = -cosec(x).

**What is cosec theta equal to?**Cosec theta of a right-angled triangle is equal to the ratio of the length of the hypotenuse to the length of the opposite side.

**In which quadrants is the cosecant function positive and in which quadrants is it negative?**

It can be observed from the above graph that cosec(x) is positive in the 1st and 2nd quadrants and negative in the 3rd and 4th quadrants.

**Know More Trigonometric Function**