How this instrument works
The three basic trigonometric functions — sine, cosine, and tangent — each measure a different ratio within a right triangle at a given angle: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent (the SOH-CAH-TOA mnemonic). This page evaluates all three at once for a single entered angle, useful for whenever a full trigonometric profile is wanted together rather than checking three separate dedicated pages one at a time.
The three values are quietly linked: tangent is always sine divided by cosine, tan(x) = sin(x)⁄cos(x), so once any two of the three are known, the third follows automatically. This page computes all three independently from the angle directly, so that link can be checked rather than assumed.
Sine and cosine both stay bounded between −1 and 1 at every angle, tracing out smooth, mirror-image waves offset by 90° from each other, while tangent is unbounded, growing without limit as the angle approaches 90°, where cosine (tangent's own denominator) shrinks toward zero.
- Enter an angle into the Angle field.
- Read sin(x), cos(x), and tan(x): the sheet computes all three simultaneously from that same angle.
- Check that tan(x) always matches sin(x) divided by cos(x).
Worked example — all three at 30°
At 30°, sin(30°) = 0.5, cos(30°) ≈ 0.866, and tan(30°) ≈ 0.577 — three genuinely different values from the same starting angle, each answering a different question about a right triangle with that angle inside it.
At 0°, sin=0, cos=1, tan=0 — the trivial reference angle. At 45°, sin and cos are equal (both ≈0.707), and their ratio, tan(45°), works out to exactly 1 — the isosceles right triangle's own defining symmetry, visible directly in these three numbers together.
Questions
What do sine, cosine, and tangent measure?
In a right triangle, sine is the side opposite an angle divided by the hypotenuse, cosine is the adjacent side divided by the hypotenuse, and tangent is the opposite side divided by the adjacent side — SOH-CAH-TOA.
How are the three functions related to each other?
Tangent always equals sine divided by cosine: tan(x) = sin(x) ⁄ cos(x). Knowing any two of the three lets you find the third directly through this relationship.
Why is tangent unbounded while sine and cosine aren't?
Sine and cosine are both coordinates on the unit circle, naturally confined between −1 and 1. Tangent is their ratio, with cosine in the denominator — as cosine shrinks toward zero near 90°, the ratio grows without any bound at all.
How is this different from the individual sine, cosine, and tangent pages?
Those pages each evaluate one function on its own. This page runs all three at once on the identical angle, for whenever the full trigonometric picture is wanted together rather than piece by piece.
Is this the forward or reverse operation compared to inverse trig functions?
Forward — this page takes an angle and returns ratios. The companion Inverse Trigonometric Functions page on this site runs the reverse direction, taking a ratio and returning the angle that produced it.