How to Calculate Cosecant
The cosecant function, abbreviated as csc(θ), is a trigonometric function that is defined as the reciprocal of the sine function. It is one of the six fundamental trigonometric functions and plays a crucial role in various mathematical and real-world applications.
The formula for cosecant is:
\[ \csc(\theta) = \frac{1}{\sin(\theta)} \]
Where:
- θ = The angle in radians
- sin(θ) = The sine of the angle θ
Calculation Steps
- Ensure the angle is in radians. If it's in degrees, convert it to radians using the formula: radians = degrees × (π / 180°).
- Calculate the sine of the angle using a calculator or trigonometric tables.
- Take the reciprocal of the sine value (divide 1 by the sine value).
- The result is the cosecant of the angle.
Example
Let's calculate the cosecant of 30°:
- Convert 30° to radians:
\[ 30° × \frac{\pi}{180°} = \frac{\pi}{6} \text{ radians} \]
- Calculate sin(π/6):
\[ \sin(\frac{\pi}{6}) = \frac{1}{2} \]
- Apply the cosecant formula:
\[ \csc(\frac{\pi}{6}) = \frac{1}{\sin(\frac{\pi}{6})} = \frac{1}{\frac{1}{2}} = 2 \]
Therefore, csc(30°) = 2.
Visual Representation
This diagram illustrates the cosecant of 30°, which is equal to 2. The blue arc represents the angle, and the vertical line represents the cosecant value.