How this instrument works
This sheet is the three-function convenience version of the standalone sine, cosine, and tangent calculators: rather than asking for one ratio at a time, it takes an angle θ once and returns all three together, since most real problems need more than one of them in the same step — resolving a force into horizontal and vertical components calls for cos θ and sin θ in the same breath, and reading the resulting direction back out of those components calls for tan θ = sin θ ⁄ cos θ on top of that.
Picture the angle swept counterclockwise from the positive x-axis to a point on a circle of radius 1: cos θ is that point's x-coordinate, sin θ is its y-coordinate, and tan θ is the slope of the ray connecting the origin to the point — rise over run, or equivalently sin θ divided by cos θ. That third definition is a ratio of the other two rather than an independent measurement, which is why tan θ can be recovered from whichever sin and cos this sheet has already found instead of needing a formula of its own.
The ratio breaks down exactly where cosine vanishes. At θ = 90° and θ = 270° the point on the unit circle sits directly above or below the origin, its x-coordinate is 0, and dividing by that zero leaves tangent undefined — the sheet flags this rather than returning a false number. Sine and cosine themselves stay perfectly well-behaved at those same angles, since neither one involves a division at all.
- Enter your angle into the Angle, θ field — it opens in degrees, but the unit selector beside it also accepts radians or turns.
- Read sin(θ), cos(θ), and tan(θ) together; all three update from the single angle you typed, with no need to enter it a second or third time.
- Watch for a flagged tan(θ) result at 90° or 270° — the sheet reports tangent as undefined there rather than showing a false number.
- Use sin(θ) and cos(θ) as the vertical and horizontal components of a unit vector at that angle; scale both by a magnitude to resolve an actual force or displacement.
- Step θ from 0° through 360° and watch sin(θ) and cos(θ) each trace one full wave while tan(θ) repeats twice as often, with a break at every 90°.
Worked example — θ = 45° gives all three at once
Set Angle, θ to 45, the field's default unit of degrees. Internally the sheet converts that to 0.7853981633974483 radians — exactly π ⁄ 4 — before evaluating, and the three outputs land at sin(θ) = 0.7071067811865476, cos(θ) = 0.7071067811865476, and tan(θ) = 0.9999999999999999, which the display rounds to a clean 1.00000000 at eight decimal places.
Forty-five degrees is the one angle where sine and cosine must agree, since it splits a right angle exactly in half and the two legs of the resulting triangle come out equal by symmetry — so seeing sin(θ) and cos(θ) return identical figures is itself a check that the sheet is behaving. Dividing either one by itself gives tan(θ) = 1 immediately, the same shortcut a 45-45-90 triangle offers by hand: equal legs mean a slope of exactly one, a 45° incline, before any trig ratio gets computed at all.
Questions
Why does this calculator return three values instead of one?
Because most uses of trigonometry need more than one ratio from the same angle at once — resolving a vector into horizontal and vertical components takes cos(θ) and sin(θ) together, and recovering the vector's direction afterward takes tan(θ) on top. Typing the angle once and reading all three off together removes the risk of re-typing it wrong a second or third time and keeps the three results internally consistent.
How is tan(θ) related to sin(θ) and cos(θ)?
tan(θ) is defined as their ratio, sin(θ) ⁄ cos(θ), not measured independently. Geometrically it is the slope of the ray from the origin to the point (cos θ, sin θ) on the unit circle: rise over run. Because it is a quotient rather than a coordinate, tan(θ) can grow without bound as cos(θ) shrinks toward zero, unlike sine and cosine, which never leave the range −1 to 1.
Why is tan(θ) undefined at 90° and 270°?
At those two angles the point on the unit circle sits directly above or below the origin, so cos(θ) equals 0 and the ratio sin(θ) ⁄ cos(θ) divides by zero. Sine and cosine themselves stay perfectly defined there — sin(90°) = 1 and cos(90°) = 0 are ordinary numbers — it is only their quotient that breaks down, which is why this sheet flags the result instead of guessing at one.
What's a quick way to check the three results by hand?
Check the Pythagorean identity: sin(θ) squared plus cos(θ) squared should always equal 1, since both are coordinates on a unit circle. At θ = 45°, squaring 0.7071067811865476 twice and adding gives 1.0000000000000002 — a floating-point smear that rounds to a clean 1 at any sane display precision. Then confirm tan(θ) by dividing the two: 0.7071067811865476 ⁄ 0.7071067811865476 = 1, matching the sheet's third output with no separate lookup needed.
Do sin(θ), cos(θ), and tan(θ) share the same angle unit?
Yes — whichever unit you pick in the Angle, θ selector, degrees, radians, or turns, applies to all three outputs at once, since they are computed from one converted angle internally. Switching the unit changes what the typed number means but never desynchronizes sine, cosine, and tangent from one another.
How does this differ from using separate sine, cosine, and tangent calculators?
The arithmetic is identical — each output here matches exactly what a standalone sine, cosine, or tangent sheet would return for the same angle. The difference is that this one evaluates θ a single time and reports all three together, which matters most when a problem, like resolving a force into components and then reading the resulting angle back out, genuinely needs more than one of the three ratios at the same time.