How this instrument works
The Pythagorean trigonometric identity states that sin²θ + cos²θ = 1 for every angle θ, without exception. It is not a coincidence or an approximation — it is the Pythagorean theorem itself, restated for the unit circle. Any point on the unit circle at angle θ sits at coordinates (cosθ, sinθ), and since that circle has radius exactly 1, the Pythagorean theorem applied to the right triangle formed by that point and the two axes gives cos²θ + sin²θ = 1² = 1 directly.
This calculator computes sin²θ and cos²θ separately for a chosen angle and adds them, so the identity can be checked against a live, arbitrary number rather than taken purely on faith. The two individual terms change constantly as the angle sweeps around the circle — sin² grows while cos² shrinks, and vice versa — but their sum never wavers from exactly 1.
The identity is the foundation nearly every other trigonometric identity is built from. Dividing straight through by cos²θ gives 1 + tan²θ = sec²θ; dividing by sin²θ instead gives 1 + cot²θ = csc²θ. Recognizing when an expression secretly contains sin²θ + cos²θ, so it can be replaced by the plain number 1, is one of the single most useful simplification tricks in trigonometry and calculus alike.
- Enter any angle into the Angle field, in whichever unit is convenient.
- Read sin²(angle) and cos²(angle): the two squared trig values computed independently for that angle.
- Read sin²(angle) + cos²(angle): the sheet adds the two together, and the result will always land on exactly 1, confirming the identity for your chosen angle.
- Try a range of very different angles — near 0°, near 90°, or somewhere in between — to see the individual terms shift dramatically while their sum stays fixed.
Worked example — a 37° angle
At 37°, sin²(37°) ≈ 0.3622 and cos²(37°) ≈ 0.6378 — two numbers that look unrelated to each other at first glance. Added together, sin²(37°) + cos²(37°) = 0.3622 + 0.6378 = 1.0 exactly, confirming the identity holds at this ordinary, unremarkable angle just as firmly as at any famous special angle.
Compare the two extremes: at 0°, sin²=0 and cos²=1 (the two terms sum to 1 trivially, all the 'weight' sitting on cosine); at 90°, sin²=1 and cos²=0 (the weight has fully swapped to sine). The individual terms trade places entirely across that quarter turn, yet the sum stays fixed at exactly 1 throughout.
Questions
What is the Pythagorean trigonometric identity?
sin²θ + cos²θ = 1, true for every angle θ. It is a direct restatement of the Pythagorean theorem applied to the unit circle: any point (cosθ, sinθ) on a circle of radius 1 satisfies x² + y² = 1 by the circle's own definition, which is exactly this identity written in trig notation.
Why does this identity matter so much?
Nearly every other trigonometric identity is derived from it — dividing through by cos²θ or sin²θ produces the other two Pythagorean identities (1 + tan²θ = sec²θ and 1 + cot²θ = csc²θ), and recognizing a hidden sin²θ + cos²θ inside a larger expression is one of the most common simplification moves in trigonometry and calculus.
Does the identity hold for negative or very large angles?
Yes, without exception. Sine and cosine are periodic and well-defined for any real-number angle, positive, negative, or larger than a full turn, and the identity sin²θ + cos²θ = 1 holds at every single one of those angles with no special cases to worry about.
How is this different from the unit circle page?
The unit circle page reports the actual coordinates (cosθ, sinθ) for a chosen angle — where the point sits. This page instead squares and adds those two values together to demonstrate that their sum is always exactly 1, proving the identity rather than just locating a point.
Can floating-point rounding make the sum slightly off from 1?
In principle, computer arithmetic can leave a sum like 0.9999999999999999 instead of a perfectly clean 1.0 due to how decimal fractions are stored in binary — a tiny rounding artifact, not a flaw in the identity itself, and far too small to matter for any practical use.