What Is the Cotangent Calculator?
The cotangent calculator finds the trigonometric cotangent of an angle, written \(\cot(\theta)\). Cotangent is one of the six standard trigonometric functions and is defined as the ratio of the cosine of an angle to its sine — or equivalently, the reciprocal of the tangent. It is used widely in trigonometry, physics, engineering, and geometry to describe relationships between sides and angles.
How to Use It
Enter the angle, then choose whether the value is in degrees or radians. The calculator converts degrees to radians internally, then computes \(\cot(\theta) = \cos(\theta)/\sin(\theta)\). The result is shown along with the angle expressed in radians for reference. Note that cotangent is undefined when \(\sin(\theta) = 0\) — that happens at 0°, 180°, 360°, and every multiple of 180°.
The Formula Explained
The core identity is \(\cot(\theta) = \cos(\theta)/\sin(\theta) = 1/\tan(\theta)\). Because the unit-circle sine appears in the denominator, the function grows without bound as the angle approaches a multiple of 180° and the sine approaches zero. For all other angles the ratio gives a finite, exact value.
$$\cot(\theta) = \frac{\cos\left(\text{Angle} \cdot \frac{\pi}{180}\right)}{\sin\left(\text{Angle} \cdot \frac{\pi}{180}\right)}$$
Worked Example
For \(\theta = 45°\): \(\cos(45°) \approx 0.70710678\) and \(\sin(45°) \approx 0.70710678\), so $$\cot(45°) = \frac{0.70710678}{0.70710678} = 1$$ For \(\theta = 30°\): \(\cos(30°) \approx 0.8660254\), \(\sin(30°) = 0.5\), so $$\cot(30°) = \frac{0.8660254}{0.5} \approx 1.7320508$$ (which is \(\sqrt{3}\)).
FAQ
What is \(\cot(90°)\)? Since \(\cos(90°) = 0\) and \(\sin(90°) = 1\), \(\cot(90°) = 0\).
Why is cotangent sometimes undefined? When \(\sin(\theta) = 0\) (at 0°, 180°, 360°…), dividing by zero is undefined, so cotangent has no value at those angles.
Is cot the same as 1/tan? Yes — cotangent is the reciprocal of tangent wherever tangent is defined and nonzero.
