How this instrument works
Sine, cosine, and tangent are the three core ratios built from a right triangle's sides, and each one is simply a function of a single angle — feed in any angle and each ratio comes back as one specific number. This sheet accepts any angle whatsoever, not just a handful of tidy ones, and reports all three ratios together rather than making a separate trip for each.
A sibling page on this site, built around the exact value of trig functions, covers only five standard angles — 0°, 30°, 45°, 60°, and 90° — and returns the first two of those ratios as clean, non-decimal values like one-half or root-three-over-two, worth memorizing since they recur so often. This page takes the opposite approach: any angle at all is fair game, the answers come back as rounded decimal approximations rather than exact fractions or roots, and a third ratio joins them, rather than being left out.
Tangent itself is nothing more than sine divided by cosine, so once those two ratios are known for a given angle, it follows for free. That also explains why it misbehaves at 90°, where cosine drops to zero: dividing by zero sends it toward infinity, a genuine discontinuity rather than a rounding quirk.
- Enter any angle into the Angle field, in degrees by default.
- Read sin(angle), cos(angle), and tan(angle) together, all three computed from the identical input.
- Switch the angle's unit to radians or turns if that suits your source data better.
- Compare two angles that add to 90°, like 30° and 60°, to watch sine and cosine trade places.
Worked example — 30° and 60° compared
At the golden case of 30°, sine comes back as exactly 0.5, cosine as roughly 0.8660254, and tangent — the first divided by the second — as roughly 0.5773503. At 0°, all three ratios simplify further still: sine is 0, cosine is 1, and the last one, being 0 divided by 1, is also 0.
Push the angle to 60° instead and notice the swap: sine now reads roughly 0.8660254, the same figure cosine held at 30°, while cosine drops to exactly 0.5, matching sine's earlier value. Tangent at 60° comes out to roughly 1.7320508 — the reciprocal of its own 30° result, since 30° and 60° are complementary angles that add to a right angle.
Questions
What three values does this calculator return?
Sine, cosine, and tangent, all evaluated at whatever angle you enter, each shown as a decimal approximation rather than an exact fraction or root. Entering 30°, for instance, returns sin=0.5, cos near 0.866, and tan near 0.577 together in one pass.
How is this different from the exact value of trig functions page?
That page restricts its input to just five special angles — 0°, 30°, 45°, 60°, and 90° — and returns exact, non-decimal values for sine and cosine only. This page accepts any angle at all, returns rounded decimal approximations instead of exact values, and adds tangent as a third result.
Why is tangent left undefined at 90°?
Because tangent equals sine divided by cosine, and the second of those drops to exactly 0 at 90° — dividing by zero has no finite answer, so it shoots toward infinity rather than settling on a number, a genuine mathematical discontinuity rather than a display error.
Do sine and cosine ever return the same value?
Yes, at 45°, where the two both land near 0.7071 — and more generally, whenever two angles are complementary and add to 90°, sine of one equals cosine of the other, which is why 30° and 60° swap values between them.
Can I enter an angle in radians instead of degrees?
Yes — switch the angle's unit selector to radians or turns, and enter the value in that unit; the same three ratios come back regardless of which unit the angle was read in, since the underlying angle stays identical either way.
Why do sine and tangent both read 0 at a 0° angle?
At 0°, the triangle has collapsed flat with no height at all — sine, which measures a triangle's height relative to its hypotenuse, is naturally 0, and the third ratio, being that value divided by cosine's own reading of 1, inherits the same 0.