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The Pythagorean Identity
sin²(θ) + cos²(θ) = 1 is true for every angle θ, and comes directly from Pythagoras' theorem.
Where It Comes From
On the unit circle, the point at angle θ is (cos(θ), sin(θ)), at distance from the centre (since the radius is ). Pythagoras' theorem on that right-angled triangle gives cos(θ)² + sin(θ)² = — the identity is really Pythagoras' theorem in disguise.
Example 1 — checking the identity
Verify the identity for θ = °.
- sin(°) ≈ , cos(°) ≈
- Square each: ≈ , and ≈
- Add: ✓
Example 2 — finding cos from sin
If sin(θ) = and θ is in Quadrant , find cos(θ).
- Use the identity: cos²(θ) = − sin²(θ) =
- Take the square root: cos(θ) = √ (positive, since θ is in Quadrant , where cos is positive)
Answer: cos(θ) =
Practice
If cos(θ) = and θ is in Quadrant , find sin(θ) to decimal places.
Your answer
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