Why sin^2 + cos^2 = 1
A point travels around the unit circle while its horizontal and vertical projections trace cosine and sine values in real time. A right triangle formed by the radius and these two legs shows why the hypotenuse always equals 1, so sin^2(theta) + cos^2(theta) = 1 holds for every angle. Useful for students first meeting trigonometric identities and for teachers linking the unit circle to right-triangle definitions.
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The prompt that made it
A point moves around the unit circle; its horizontal and vertical legs are cos and sin of the angle, the radius is 1, and Pythagoras gives sin^2 + cos^2 = 1 for every angle.
Make the next one in this series
- Show how tan(theta) appears as a tangent line segment on the unit circle diagram.
- Animate the graphs of sin(theta) and cos(theta) being traced out as the point moves around the circle.
- Demonstrate the angle addition formulas using two rotating points on the unit circle.
