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Trigonometric Identities - Theory & Concepts - Trigonometry Double Angle Sum

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

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Angle Sum Geometric Proof

Explore the geometric proof of the identity sin⁡(α+β)=sin⁡αcos⁡β+cos⁡αsin⁡β\sin(\alpha+\beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta.

Angle α\alpha30°
10°45°
Angle β\beta25°
10°45°

Geometric Breakdown

Lower Segment (TR=QSTR = QS):
QS=sin⁡αcos⁡β=sin⁡(30∘)cos⁡(25∘)≈0.4532QS = \sin\alpha\cos\beta = \sin(30^\circ)\cos(25^\circ) \approx 0.4532
Upper Segment (PTPT):
PT=cos⁡αsin⁡β=cos⁡(30∘)sin⁡(25∘)≈0.3660PT = \cos\alpha\sin\beta = \cos(30^\circ)\sin(25^\circ) \approx 0.3660
Total Height (PR=sin⁡(α+β)PR = \sin(\alpha+\beta)):
sin⁡(30∘+25∘)=sin⁡(55∘)≈0.8192\sin(30^\circ+25^\circ) = \sin(55^\circ) \approx 0.8192
Sum of segments: 0.4532+0.3660=0.81920.4532 + 0.3660 = 0.8192
In right triangle △OPQ\triangle OPQ, hypotenuse OP=1OP = 1. Projection of PP onto the horizontal gives vertical height PR=sin⁡(α+β)PR = \sin(\alpha+\beta). This partition splits PRPR into PTPT and TRTR.
Geometric proof for double angle sum trigonometric identityX AxisY AxisUnit Circle BoundarySegment OQ (length = cos beta)Segment QP (length = sin beta)Segment OP (length = 1)Total Height PR (sin(alpha + beta))Vertical Height QS (cos beta * sin alpha)Horizontal line QTPoint OPoint PPoint QPoint RPoint SPoint TOPQRSTSegment TR (sin alpha * cos beta)Segment PT (cos alpha * sin beta)Angle alpha arcαAngle beta arcβAngle alpha at vertex Pα
PT = cos α · sin βTR = sin α · cos βPR = sin(α+β)

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