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Calculate the tangent of any angle in degrees or radians. Tangent is the ratio of sine to cosine (tan = sin/cos).
| Degrees | Radians | Exact Value | Decimal |
|---|---|---|---|
| 0° | 0 | 0 | 0.0000 |
| 30° | π/6 | √3/3 | 0.5774 |
| 45° | π/4 | 1 | 1.0000 |
| 60° | π/3 | √3 | 1.7321 |
| 90° | π/2 | undefined | — |
| 120° | 2π/3 | -√3 | -1.7321 |
| 135° | 3π/4 | -1 | -1.0000 |
| 150° | 5π/6 | -√3/3 | -0.5774 |
| 180° | π | 0 | -0.0000 |
| 270° | 3π/2 | undefined | — |
Domain: All θ except 90° + n·180°
Range: All real numbers (-∞, ∞)
Period: π radians (180°)
tan(θ + π) = tan(θ)
Odd function: tan(-θ) = -tan(θ)
Origin symmetry
At θ = 90°, 270°, ...
Where cos(θ) = 0
The tangent function is one of the six fundamental trigonometric functions. It is defined as the ratio of sine to cosine: tan(θ) = sin(θ)/cos(θ). In a right triangle, tangent equals the ratio of the opposite side to the adjacent side.
On the unit circle, tangent can be visualized as the length of the line segment from the origin to the point where the terminal side of the angle intersects the vertical line x = 1. This geometric interpretation explains why tangent has vertical asymptotes at 90° and 270°, where the terminal side becomes vertical and never intersects the line x = 1.
Because tan = sin/cos, and cos(90°) = cos(270°) = 0. Division by zero is undefined.
Tangent is the length of the line segment from the origin to where the terminal side intersects the line x = 1 (the tangent line to the unit circle).
Tangent is positive in Quadrants I and III (where sin and cos have the same sign), and negative in Quadrants II and IV.
The period of tangent is π radians (180°), which is half the period of sine and cosine. This means tan(θ + π) = tan(θ).
The domain is all real numbers except odd multiples of 90° (π/2 radians), where cosine equals zero.
The range is all real numbers (-∞, ∞). Unlike sine and cosine, tangent is unbounded.
The tangent of an angle equals the slope of a line making that angle with the positive x-axis. This makes tangent useful in calculating gradients and angles of inclination.
Vertical asymptotes occur where tangent is undefined (at 90°, 270°, etc.). The function approaches positive or negative infinity as it nears these angles.