Calculate the tangent of any angle in degrees or radians. Identifies vertical asymptotes, exact values for special angles, and ratios.
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In trigonometry, the tangent of an angle is the ratio of the opposite leg to the adjacent leg in a right triangle: tan(θ) = Opposite / Adjacent = sin(θ) / cos(θ). It also represents the slope of the terminal ray.
\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{\sin(\theta)}{\cos(\theta)} = \text{slope } mTangent is undefined whenever cos(θ) = 0, which occurs at odd multiples of 90° (±90°, ±270°, etc.).
Scenario: Calculate tan(45°)
Given: θ = 45°
In a 45°-45°-90° triangle, opposite leg equals adjacent leg.
tan(45°) = sin(45°) / cos(45°) = (√2/2) / (√2/2) = 1
At 90°, cos(90°) = 0. Because tan(θ) = sin(θ) / cos(θ), evaluating tan(90°) requires dividing by zero (1 / 0), which is mathematically undefined.
tan(0°) = 0, tan(30°) = √3/3 ≈ 0.577, tan(45°) = 1, tan(60°) = √3 ≈ 1.732.
The period of tangent is 180° (or π radians): tan(θ + π) = tan(θ).
The reciprocal of tangent is cotangent: cot(θ) = 1 / tan(θ) = cos(θ) / sin(θ).
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