Pythagorean identities
Relate squared trigonometric functions through the unit circle.
Pythagorean identities is one of 5 identities formulas in the trigonometry section of this library, and it is used at high school · ap level.
Why pythagorean identities works
A point on the unit circle has coordinates (cos θ, sin θ) and sits exactly 1 unit from the center, so x² + y² = 1 becomes cos²θ + sin²θ = 1. Divide that whole equation by cos²θ and every term turns into the tangent version; divide by sin²θ instead and you get the cotangent version. All three are one fact wearing three outfits.
What each symbol means
$\theta$ is an angle where the displayed functions exist.
Pythagorean identities: when it holds
When taking square roots, choose sign from the angle’s quadrant.
When it stops applying
Taking a square root at the end throws the sign away. If sin θ = 3/5 you get cos²θ = 16/25 and cos θ = ±4/5, and the identity cannot tell you which. In quadrant I the answer is 4/5, but in quadrant II the same sine goes with cos θ = −4/5.
Pythagorean identities: a worked example
If $\sin\theta=3/5$ in quadrant I, then $\cos\theta=4/5$.
The mistake to avoid
What people do: Students read sin²θ as sin(θ²) and square the angle before pressing the sine key.
Why it goes wrong: The exponent sits on the output, not the input. sin²θ is shorthand for (sin θ) times (sin θ), and squaring an angle first gives a totally unrelated number.
Do this instead: Say it out loud as sine of theta, squared. On a calculator: take the sine, then square the answer. With θ = 37 degrees the sine is about 0.6, so sin²θ is about 0.36.
Pythagorean identities: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $\theta$ is an angle where the displayed functions exist.
- Check the conditions before substituting. When taking square roots, choose sign from the angle’s quadrant.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most trigonometry slips.
Where this formula fits
- Subject
- Trigonometry formulas — 18 entries in this library
- Topic
- Identities
- Level
- High school · AP
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where pythagorean identities comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Trigonometric Identities — the lesson behind this formula: simplify and prove relationships between trig functions.
- Trigonometry Calculator — check your substitution and the value it produces.
- Study trigonometry — the subject guide that explains the ideas these formulas compress.
- Precalculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 18 trigonometry formulas — the full grouped reference, or the complete formula library.
Questions about pythagorean identities
Do I have to memorize all three versions?
No, one is enough. Start from sin²θ + cos²θ = 1 and divide every term by cos²θ to produce 1 + tan²θ = sec²θ on the spot. That takes about ten seconds and removes two things from your memory list.
Does the identity still hold when the angle is in degrees?
Yes. The equation is about the position of a point on the circle, and switching units only changes how you name the angle. Just keep your calculator in one mode so the two functions describe the same point.
How do I decide the sign after taking a square root?
Find the quadrant of the angle first, then use it: cosine is negative in quadrants II and III, and sine is negative in quadrants III and IV. The quadrant chooses the sign, and the identity supplies only the size.
Is this really just the Pythagorean theorem again?
It is the same theorem applied to the little right triangle under the unit-circle point, whose legs measure cos θ and sin θ and whose hypotenuse is the radius 1. The familiar a² + b² = c² becomes cos²θ + sin²θ = 1.