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Trigonometric Identities - Theory & Concepts - Trigonometry Pythagorean Identity

Fundamental identities, Pythagorean identities, Sum/Difference, Double/Half Angle formulas, Co-function identities, Power-Reducing formulas, and proofs.

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Pythagorean Trigonometric Identity

Visualize the classic identity sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1 geometrically. Adjust the angle to see how the squares scale while maintaining their sum.

Angle (θ\theta)36°
0°30°45°60°90°

Geometric Areas

Areaadj=cos⁡2θ\text{Area}_{\text{adj}} = \cos^2\theta0.8090^2 = 0.6545
Areaopp=sin⁡2θ\text{Area}_{\text{opp}} = \sin^2\theta0.5878^2 = 0.3455
Areahyp=12\text{Area}_{\text{hyp}} = 1^21.0000
Sum of Areas:0.6545 + 0.3455 = 1.0000
By the Pythagorean Theorem, in any right triangle with hypotenuse c=1c=1 and legs a=sin⁡θa=\sin\theta, b=cos⁡θb=\cos\theta:
a2+b2=c2  ⟹  sin⁡2θ+cos⁡2θ=1a^2 + b^2 = c^2 \implies \sin^2\theta + \cos^2\theta = 1
Pythagorean right triangle identity simulationHorizontal AxisVertical AxisUnit Circle Quarter ArcCosine Squared Square AreaSine Squared Square AreaHypotenuse Squared Square AreaRight TriangleRight Angle IndicatorTerminal Point on Unit Circlecos θsin θ1
Hypotenuse Square (Area = 1.00)

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