Trig Identities Lab: sin²θ + cos²θ = 1
Skill goal: See and prove the Pythagorean identity on the unit circle: the two legs always build squares whose areas add to 1.
1 · Observe
Drag the angle slider. The blue square has area sin²θ, the teal square has area cos²θ.
2 · Manipulate
3 · Predict
Before moving the slider: sin²(30°) + cos²(30°) = ? Now drag θ to 135°. Does the sum change? Why or why not?
4 · See
5 · Explain
On the unit circle the hypotenuse is always 1. By the Pythagorean theorem, (cosθ)² + (sinθ)² = 1². The two colored squares are built on the legs of the right triangle, so their areas must always add to exactly 1 — for every angle.
6 · Challenge
Q: If sinθ = 0.6 and θ is in Quadrant I, then cosθ = ?