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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 1 from the centre (since the radius is 1). Pythagoras' theorem on that right-angled triangle gives cos(θ)² + sin(θ)² = 1^{2} = 1 — the identity is really Pythagoras' theorem in disguise.

Unit circle

Angle 40°

cos(40°) ≈ 0.766 · sin(40°) ≈ 0.643 · point ≈ (0.766, 0.643)

Change a value and watch:

Watch cos(θ) and sin(θ). Mentally square and add — you should always get about 1 .

The radius is 1, so the right triangle from the origin to (cos θ, sin θ) has hypotenuse 1 .

Example 1 — checking the identity

Verify the identity for θ = 45 °. 1 sin(45 °) \approx 0.707, cos(45 °) \approx 0.707 2 Square each: 0.70 7^{2} \approx 0.5, and 0.70 7^{2} \approx 0.5 3 Add: 0.5 + 0.5 = 1 ✓

Example 2 — finding cos from sin

If sin(θ) = 0.6 and θ is in Quadrant 1, find cos(θ). 1 Use the identity: cos²(θ) = 1 - sin²(θ) = 1 - 0.36 = 0.64 2 Take the square root: cos(θ) = \sqrt 0.64 = 0.8 (positive, since θ is in Quadrant 1, where cos is positive) Answer: cos(θ) = 0.8

Practice

If cos(θ) = 0.28 and θ is in Quadrant 1, find sin(θ) to 2 decimal places.

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