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
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).
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.
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.
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