The six trigonometric functions describe the relationship between an angle and the sides of a right triangle — or, more generally, between an angle and a point on the unit circle. This calculator handles both directions: turning an angle into all six function values, and turning a known ratio back into an angle with the inverse functions.
From right triangles to the unit circle
Trigonometry starts with SOH-CAH-TOA: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, tangent is opposite over adjacent. That definition only works for angles between 0° and 90°, though, since a right triangle can't have an angle of 180° or −45°. The unit-circle definition extends the functions to every real angle: sine is the y-coordinate and cosine is the x-coordinate of the point where the angle's ray meets a circle of radius 1, and tangent is their ratio. This is why sine and cosine are always between −1 and 1 (the circle's coordinates never exceed that range), while tangent can be any real number.
Why some functions are undefined at certain angles
Tangent, secant, and cotangent are all defined as ratios, and a ratio is undefined when its denominator is zero. Tangent (sin/cos) and secant (1/cos) break down wherever cosine equals zero — at 90°, 270°, and every 180° after that. Cotangent (cos/sin) and cosecant (1/sin) break down wherever sine equals zero — at 0°, 180°, 360°, and so on. Graphically, these are the vertical asymptotes you see on a tangent or secant curve: the function shoots toward positive or negative infinity as the angle approaches the undefined point, but never actually reaches a value there.
Why inverse trig has a restricted domain
Because sine and cosine only ever output values between −1 and 1, their inverse functions (sin⁻¹ and cos⁻¹) can only accept inputs in that same range — there's no angle whose sine is 2, so sin⁻¹(2) simply doesn't exist. Arctangent doesn't have this restriction because tangent's output range is all real numbers, so tan⁻¹ accepts any value. A second subtlety: sine and cosine are periodic, so infinitely many angles share the same sine value (30°, 150°, 390°, …). Inverse functions return exactly one of them — the "principal value" — which for sin⁻¹ is always between −90° and 90°, for cos⁻¹ between 0° and 180°, and for tan⁻¹ between −90° and 90°.
Degrees vs. radians
Degrees split a circle into 360 equal parts, a convention going back to ancient Babylonian astronomy. Radians measure angle by arc length: one radian is the angle where the arc length equals the radius, giving 2π radians per full circle. Radians are the natural unit in calculus and physics because they make derivative and series formulas for sine and cosine simplify cleanly, while degrees remain the everyday unit for navigation, construction, and most textbook geometry. This calculator accepts and displays both, so you can work in whichever unit your source material uses.