The Pythagorean Identity
Simple Explanation
For any angle θ, sin²θ + cos²θ = 1 — always, with no exceptions. This is called the Pythagorean identity because it comes directly from the Pythagorean theorem applied to the general trig definition.
Why Do We Need It?
This identity is one of the most-used tools in trigonometry — it lets you find sin θ from cos θ (or vice versa), and simplifies countless later trigonometric expressions and proofs.
See It
A unit circle with a point P at (0.6, 0.8), showing the right triangle formed by its x and y components, illustrating that 0.6 squared plus 0.8 squared equals 1
Formula
The Pythagorean Identity
sin²θ + cos²θ = 1
For any angle θ, the square of its sine plus the square of its cosine always equals exactly 1.
- θ
- — any angle
When to use it: Whenever one of sin θ or cos θ is known and the other is needed, or to simplify an expression containing both.
Worked Example
Find cos θ given sin θ, using the Pythagorean identity
Given sin θ = 0.6 and θ is in Quadrant I, find cos θ.
Why Does This Work?
On the unit circle (r=1), the point (cos θ, sin θ) lies at distance exactly 1 from the origin — and by the Pythagorean theorem, the distance from the origin to any point (x,y) is √(x²+y²). Setting √(cos²θ+sin²θ) = 1 and squaring both sides gives sin²θ+cos²θ=1 directly.
Real-Life Example
Checking a resolved vector's components
An engineer resolves a unit-length force vector into its horizontal and vertical components using an angle.
The Pythagorean identity gives an instant consistency check: the squares of the two components must always sum to exactly 1 for a unit vector.
Practice
Given cos θ = −0.28 and θ is in Quadrant III (where sine is negative), find sin θ. (Give your answer to 2 decimal places.)
MediumCommon mistake
Forgetting to check the quadrant when taking the square root — √(cos²θ) could be +cos θ or −cos θ, and only the quadrant tells you which sign is actually correct.
Quick Review
- sin²θ + cos²θ = 1, for every angle θ.
- Comes directly from the Pythagorean theorem on the unit circle.
- Use the quadrant to choose the correct sign after taking a square root.