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Business Mathematics and Basic Statistics · Ch 6 — Trigonometry

Complementary and Supplementary Angle Relations

4

Complementary and Supplementary Angle Relations

Two angles are complementary if they add up to 90°90°, and supplementary if they add up to 180°180°. Both relationships give useful shortcuts for rewriting one trigonometric ratio in terms of another, without any new calculation.

Complementary angles (θ\theta and 90°−θ90°-\theta): in a right triangle, the two acute angles are always complementary, and the side that is "opposite" one of them is "adjacent" to the other. Swapping opposite and adjacent swaps sine and cosine (and tangent and cotangent, cosecant and secant):

sin⁡(90°−θ)=cos⁡θcos⁡(90°−θ)=sin⁡θtan⁡(90°−θ)=cot⁡θ\sin(90°-\theta) = \cos\theta \qquad \cos(90°-\theta) = \sin\theta \qquad \tan(90°-\theta) = \cot\theta

cot⁡(90°−θ)=tan⁡θsec⁡(90°−θ)=cosec θcosec(90°−θ)=sec⁡θ\cot(90°-\theta) = \tan\theta \qquad \sec(90°-\theta) = \text{cosec}\,\theta \qquad \text{cosec}(90°-\theta) = \sec\theta

Supplementary angles (θ\theta and 180°−θ180°-\theta): 180°−θ180°-\theta lies in the second quadrant, where sine stays positive but cosine and tangent become negative:

sin⁡(180°−θ)=sin⁡θcos⁡(180°−θ)=−cos⁡θtan⁡(180°−θ)=−tan⁡θ\sin(180°-\theta) = \sin\theta \qquad \cos(180°-\theta) = -\cos\theta \qquad \tan(180°-\theta) = -\tan\theta

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