Tangent Calculator
Type or scrub the slider to evaluate tan(θ) instantly. Exact values appear automatically for the canonical angles (0°, 30°, 45°, 60°, 90°, …); decimal values everywhere else. The unit-circle visual updates live, and the tangent curve below shows where the function blows up at θ = 90° + 180°·k.
Angle
Updates as you typeKeyboard: focus the input then ←/→ to nudge 1°, Shift+←/→ for 15°. D/R/G switch units.
Tangent curve
y = tan(θ) — vertical asymptotes at ±90°, ±270°The tangent function repeats every 180° (period π) and shoots to ±∞ at every 90° + 180°·k. The grey vertical lines mark those asymptotes; near them, tiny shifts in θ cause huge jumps in tan(θ).
Show the working
Common angles reference
Tap a row to load it · current angle highlighted| Deg | Rad | sin | cos | tan | cot |
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Formula
- θ
- The angle. Degrees, radians, or gradians — the calculator normalises both ways so you can type either.
- Opposite
- The right-triangle side facing θ.
- Adjacent
- The right-triangle side touching θ that is not the hypotenuse.
- Reference angle
- The acute angle between the terminal side and the nearest x-axis. tan(θ) and tan(reference) share the same magnitude; the sign is set by the quadrant.
- Quadrant signs
- tan is positive in quadrants I and III, negative in II and IV.
- Angle θ = —
- Convert to radians = —
- Quadrant = — → sign of tan is —
- sin(θ) = —, cos(θ) = —
- tan(θ) = sin / cos = —
- tan(θ) = —
When θ is one of the canonical angles (0°, 30°, 45°, 60°, …), the calculator shows the exact value (e.g. √3, √3/3). Otherwise it falls back to a high-precision decimal.
Examples
How It Works
Because cos(θ) appears in the denominator, tan(θ) is undefined whenever cos(θ) = 0 — at θ = 90°, 270°, 450°, … (or π/2 + π·k in radians). The graph of y = tan(θ) has a vertical asymptote at each of these angles: as θ approaches them, tan(θ) shoots to +∞ from one side and −∞ from the other.
The function is periodic with period 180° (π radians), so tan(θ + 180°) = tan(θ). It is also odd, meaning tan(−θ) = −tan(θ). Combined, these properties mean that knowing tan(θ) on the interval (−90°, 90°) is enough to know it everywhere else — every other input is just a shifted or reflected copy.
For exam and textbook problems, the canonical angles 0°, 30°, 45°, 60°, 90°, 120°, 135°, 150°, 180°, … all have exact tangent values built from √2 and √3. The calculator shows those exact forms automatically (e.g. tan(60°) = √3, tan(30°) = √3/3); for any other angle it falls back to a high-precision decimal.
Tips & Best Practices
Frequently Asked Questions
Why is tan(90°) undefined?
Because tan(θ) = sin(θ) / cos(θ) and cos(90°) = 0. Division by zero has no defined value, so tan(90°) is left undefined. The same happens at every angle of the form 90° + 180°·k.
How do I convert between degrees and radians?
Multiply degrees by π/180 to get radians, or radians by 180/π to get degrees. So 45° = π/4 ≈ 0.785 rad, and 1 rad ≈ 57.296°.
What is the period of tan?
tan has a period of 180° (π radians) — half the period of sine and cosine. That means tan(45°) = tan(225°) = tan(405°) = 1, and so on.
When should I use radians vs degrees?
Use degrees for geometry, surveying, navigation, and most everyday measurement. Use radians for calculus, physics, engineering, and any context where derivatives or arc lengths matter — d/dx[tan(x)] = sec²(x) only when x is in radians.
What is a gradian?
A gradian (or gon) divides a right angle into 100 parts, so a full circle is 400 grad. It is mostly used in surveying. The calculator supports it via the GRAD toggle.
Why does the unit circle help?
Drawing the angle on a circle of radius 1 makes the trig values geometric: cos(θ) is the x-coordinate of the point, sin(θ) is the y-coordinate, and tan(θ) is the slope of the line from the origin to that point — which is exactly y/x = sin/cos.
Is tan(−θ) the same as tan(θ)?
No — tan is an odd function: tan(−θ) = −tan(θ). For example, tan(−45°) = −1, while tan(45°) = 1. Sine is also odd, but cosine is even (cos(−θ) = cos(θ)).