This trigonometry calculator evaluates all six trig functions — sin, cos, tan, csc, sec, cot — plus their inverses, in degrees or radians, for any angle you throw at it, including the ones beyond the textbook triangle.
The Unit Circle Is the Whole Subject
For angle θ on the unit circle: cos θ is the x-coordinate, sin θ the y — triangles are just the first-quadrant shadow of that picture, which is why the functions happily accept 200°, −45° or 7π/3 here. The reciprocal trio (csc, sec, cot) are 1/sin, 1/cos, 1/tan — defined wherever their parents aren’t zero. Triangle-side solving proper lives on our Triangle Calculator. Part of our Math Calculators collection.
Worked Examples (Verified)
- sin 30° = 0.5 exactly · cos 30° = √3/2 ≈ 0.8660 · tan 30° ≈ 0.5774 — the special angles every course memorizes, confirmed on the cards (identity: 0.25 + 0.75 = 1 ✔)
- arctan(1) = 45° — where sin and cos meet, tan is exactly 1
- sin 200° ≈ −0.342 — third quadrant, below the axis: the circle explains the sign, the triangle never could
- tan 90°: undefined — the point (0, 1) has no slope; honest tools say so
The Reference Table (Degrees)
| θ | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined |
Related Tools
- Triangle Calculator — sides, angles, area together
- Complex Number Calculator — where trig meets polar form
- Math Calculators — the full collection
Frequently Asked Questions
How do I find sin, cos and tan of an angle?
On the unit circle, sin is the y-coordinate and cos the x at that angle; tan = sin/cos. Memorized anchors: sin 30° = 0.5, cos 60° = 0.5, tan 45° = 1 — this calculator evaluates any angle in degrees or radians and shows all three at once.
Degrees or radians — which should I use?
Geometry and navigation speak degrees; calculus and physics speak radians (only radians make d/dx sin x = cos x true without correction factors). The conversion: rad = deg × π/180. This tool’s toggle controls both input interpretation and inverse-function output.
Why is tan 90° undefined?
tan = sin/cos and cos 90° = 0 — division by zero. Approaching 90° from below, tan rockets to +∞; from above, −∞. Calculators that print a huge number there are showing floating-point dust; this one says “undefined.”
What are csc, sec and cot?
The reciprocals: csc = 1/sin, sec = 1/cos, cot = 1/tan. They compress formulas in calculus and older engineering texts; all six appear on the results cards here.
What range do inverse trig functions return?
The principal values: arcsin ∈ [−90°, 90°], arccos ∈ [0°, 180°], arctan ∈ (−90°, 90°). Equations like sin x = 0.5 have infinitely many solutions (30°, 150°, 390°…); the inverse function returns the representative, and solution sets are built from it with the symmetries of the circle.
Is this trigonometry calculator free?
Yes — free, instant, no sign-up.