Trigonometric Functions Calculator (sin, cos, tan)

The six functions, from the unit circle

Draw a circle of radius 1 centered at the origin. For an angle $\theta$ measured from the positive x-axis, the point where its ray crosses the circle has coordinates:

$\cos\theta = x, \qquad \sin\theta = y, \qquad \tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{y}{x}$

The other three functions are simply reciprocals:

$\cot\theta = \frac{1}{\tan\theta}, \qquad \sec\theta = \frac{1}{\cos\theta}, \qquad \csc\theta = \frac{1}{\sin\theta}$

Degrees, radians, and π multiples

This calculator accepts an angle in any of three units:

The conversion between degrees and radians:

$\text{radians} = \text{degrees} \times \frac{\pi}{180}, \qquad \text{degrees} = \text{radians} \times \frac{180}{\pi}$

So 180° = π rad, 90° = π/2, and 45° = π/4.

Worked example: 30°

$\sin 30° = \frac12 = 0.5, \qquad \cos 30° = \frac{\sqrt3}{2} \approx 0.866, \qquad \tan 30° = \frac{\sqrt3}{3} \approx 0.577$

$\csc 30° = 2, \qquad \sec 30° = \frac{2\sqrt3}{3} \approx 1.155, \qquad \cot 30° = \sqrt3 \approx 1.732$

The angle π/6 gives exactly the same result — enter 1/6 with the unit set to "π multiples".

Exact values at special angles

Every multiple of 30° or 45° in any quadrant reduces to one of these five rows, with a sign from the quadrant (see below). Any other angle only has a decimal approximation — there's no simpler radical form.

When a function is undefined

$\tan\theta$ and $\sec\theta$ divide by $\cos\theta$, so they're undefined wherever $\cos\theta = 0$ — at 90°, 270°, and every odd multiple of 90°. $\cot\theta$ and $\csc\theta$ divide by $\sin\theta$, undefined at 0°, 180°, 360°, and every multiple of 180°.

This is a genuine feature of the function, not an input error — the calculator reports "undefined" for exactly the functions that divide by zero at that angle, while the other four stay perfectly well-defined.

Periodicity and sign (ASTC)

• Period: sin, cos, sec, and csc repeat every 360° (2π); tan and cot repeat every 180° (π).
• Sign by quadrant — the classic "All Students Take Calculus" mnemonic: quadrant I, all six are positive; II, only sin (and csc); III, only tan (and cot); IV, only cos (and sec).

For example, sin 210° = -1/2: 210° sits in quadrant III (only tan/cot positive there), so sin flips negative relative to its quadrant-I reference angle of 30°.

Common mistakes

• Wrong angle mode. Typing 30 while the unit is set to radians evaluates sin/cos/tan of 30 radians (over 1700°), not 30°.
• Calling tan or sec at 90°. These genuinely have no value there — check for "undefined" rather than expecting a huge number.
• Forgetting to convert a π-multiple. 2 in π-multiple mode means 2π (360°), not the angle 2°.

Where this shows up

• Oscillation and waves — sound, light, and AC electrical signals are all modeled with sin/cos of time.
• Rotation and graphics — any 2D rotation or angle calculation ultimately calls these six functions.
• Solving general triangles — the Law of Cosines and Law of Sines extend right-triangle trigonometry to any triangle, acute or obtuse.

This calculator generalizes Trigonometric Ratios (SOH-CAH-TOA, limited to 0°–90° from two triangle sides) to every angle via the unit circle. It pairs naturally with Right Triangle, which solves a full right triangle's sides and angles directly.