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The Cotangent Calculator computes the cotangent of any angle in degrees or radians. Cotangent (abbreviated cot) is the reciprocal of tangent and can be expressed as the ratio of cosine to sine, making it the sixth and final standard trigonometric function.
Cotangent is defined as:
$$\cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta} = \frac{\text{adjacent}}{\text{opposite}}$$
Cotangent is undefined wherever $$\sin\theta = 0$$, which occurs at $$\theta = 0°, 180°, 360°,$$ and all integer multiples of $$180°$$. At these angles, the function has vertical asymptotes. This is exactly the opposite of tangent, which has asymptotes where cosine is zero.
Like tangent, cotangent is unbounded — it can take any real value. It has a period of $$\pi$$ radians (180°): $$\cot(\theta + 180°) = \cot\theta$$. It is an odd function: $$\cot(-\theta) = -\cot\theta$$.
Key exact values include:
$$\cot 30° = \sqrt{3}, \quad \cot 45° = 1, \quad \cot 60° = \frac{1}{\sqrt{3}}, \quad \cot 90° = 0$$
Notice that cotangent values are the reverse of tangent values at complementary angles: $$\cot\theta = \tan(90° - \theta)$$. This co-function relationship gives cotangent its name — it is the "complementary tangent."
Cotangent satisfies the Pythagorean identity:
$$1 + \cot^2\theta = \csc^2\theta$$
This identity is frequently used in trigonometric simplifications and integral evaluations. It directly yields the derivative of cotangent: $$\frac{d}{dx}\cot(x) = -\csc^2(x)$$.
In coordinate geometry, while tangent gives the slope of a line, cotangent gives the reciprocal of the slope, which is sometimes more convenient. If a line rises at angle $$\theta$$, its cotangent equals the horizontal run per unit of vertical rise.
Historically, cotangent was as commonly used as tangent in navigation, surveying, and astronomical calculations. Logarithmic tables of cotangent values were published alongside those of the other five functions. In modern computer science, cotangent appears in 3D graphics for constructing projection matrices, where the field-of-view angle determines the projection scaling factor via cotangent.
In complex analysis, the cotangent function has the elegant partial fraction expansion: $$\pi\cot(\pi z) = \frac{1}{z} + \sum_{n=1}^{\infty}\left(\frac{1}{z-n} + \frac{1}{z+n}\right)$$, which connects it to number theory and the Riemann zeta function.
This calculator displays both cotangent and tangent values side by side, allowing direct verification of the reciprocal relationship $$\cot\theta \times \tan\theta = 1$$ (when both are defined).
The calculator converts the angle to radians if necessary, then computes $$\cot\theta = \frac{\cos\theta}{\sin\theta}$$. If $$|\sin\theta| < 10^{-10}$$, cotangent is undefined (NaN). Tangent is computed similarly, undefined when $$|\cos\theta| < 10^{-10}$$.
Cotangent can be any real number. Positive values occur in Quadrants I and III, negative in Quadrants II and IV — the same sign pattern as tangent. An undefined cotangent means sine is zero; an undefined tangent means cosine is zero. When both are defined, their product equals 1.
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cot(45°) = cos(45°)/sin(45°) = 1. At 45°, the adjacent and opposite sides are equal, so both tangent and cotangent equal 1. This is the only angle where cot = tan.
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cot(30°) = √3 ≈ 1.7321, which is the reciprocal of tan(30°) = 1/√3 ≈ 0.5774. Notice that cot(30°) = tan(60°), confirming the co-function identity.
Since $$\cot\theta = \frac{\cos\theta}{\sin\theta}$$, it is undefined when $$\sin\theta = 0$$. This occurs at $$0°, 180°, 360°,$$ etc. At these angles, the opposite side of the triangle has zero length, making the ratio undefined. The graph has vertical asymptotes at these points.
Cotangent is the reciprocal of tangent: $$\cot\theta = \frac{1}{\tan\theta}$$. They are also co-functions: $$\cot\theta = \tan(90° - \theta)$$. When one is zero, the other is undefined, and vice versa. Their product equals 1 whenever both are defined.
Cotangent has a period of $$\pi$$ radians or 180°, the same as tangent. This means $$\cot(\theta + 180°) = \cot\theta$$ for all angles where cotangent is defined. One complete cycle of the cotangent curve spans from one asymptote to the next.
$$\frac{d}{dx}\cot(x) = -\csc^2(x)$$. The derivative is always negative (since $$\csc^2(x) > 0$$), meaning cotangent is a strictly decreasing function on each interval between its asymptotes. The integral is $$\int \cot(x)\,dx = \ln|\sin(x)| + C$$.
In computer graphics, the perspective projection matrix uses cotangent of half the field-of-view angle to determine how the 3D scene is projected onto the 2D screen. Specifically, the focal length $$f = \cot(\text{FOV}/2)$$ controls the zoom level. A narrower FOV gives a larger cotangent, producing more zoom.
The cotangent graph is a decreasing curve between consecutive asymptotes, crossing zero at $$90° + n \cdot 180°$$. It goes from $$+\infty$$ just after an asymptote to $$-\infty$$ just before the next. Unlike tangent (which increases), cotangent always decreases within each period. The graph has rotational symmetry about each zero crossing.
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