How to Calculate Cotangent
The cotangent function, abbreviated as cot(θ), is a trigonometric function that is defined as the reciprocal of the tangent function. It relates the angle of a right triangle to the ratio of the adjacent side length to the opposite side length.
The formula for cotangent is:
\[ \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\cos(\theta)}{\sin(\theta)} = \frac{\text{adjacent}}{\text{opposite}} \]
Where:
- θ = The angle in radians
- cos(θ) = The cosine of the angle
- sin(θ) = The sine of the angle
- adjacent = The length of the side adjacent to the angle in a right triangle
- opposite = The length of the side opposite to the angle in a right triangle
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 cosine and sine of the angle.
- Divide the cosine by the sine to get the cotangent.
- The result is the cotangent of the angle.
Example
Let's calculate the cotangent of 45°:
- Convert 45° to radians:
\[ 45° × \frac{\pi}{180°} = \frac{\pi}{4} \text{ radians} \]
- Calculate cos(π/4) and sin(π/4):
\[ \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \]
\[ \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \]
- Calculate cot(π/4):
\[ \cot(\frac{\pi}{4}) = \frac{\cos(\frac{\pi}{4})}{\sin(\frac{\pi}{4})} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1 \]
Therefore, cot(45°) = 1.
Visual Representation
This diagram illustrates the cotangent of 45°, which is equal to 1. The blue arc represents the angle, and the red line represents the cotangent value.