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11TH•Mathematics•Trigonometric Identities•10.2 Allied Angles

10.2 Trigonometric Ratios of Allied Angles

Trigonometric ratios of allied angles and the quadrant memory device.

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10.2 Trigonometric Ratios of Allied Angles

Angles α\alpha and β\beta are allied if α±β=n⋅90∘\alpha \pm \beta = n\cdot 90^\circ for n∈Zn \in \mathbb{Z}. Examples: ±α\pm\alpha, 90∘±α90^\circ\pm\alpha, 180∘±α180^\circ\pm\alpha, 270∘±α270^\circ\pm\alpha, 360∘±α360^\circ\pm\alpha.

Key identities:

sin⁡(π2−θ)=cos⁡θ,cos⁡(π2−θ)=sin⁡θ,tan⁡(π2−θ)=cot⁡θsin⁡(π2+θ)=cos⁡θ,cos⁡(π2+θ)=−sin⁡θ,tan⁡(π2+θ)=−cot⁡θsin⁡(π−θ)=sin⁡θ,cos⁡(π−θ)=−cos⁡θ,tan⁡(π−θ)=−tan⁡θsin⁡(π+θ)=−sin⁡θ,cos⁡(π+θ)=−cos⁡θ,tan⁡(π+θ)=tan⁡θsin⁡(3π2−θ)=−cos⁡θ,cos⁡(3π2−θ)=−sin⁡θsin⁡(2π−θ)=−sin⁡θ,cos⁡(2π−θ)=cos⁡θ,tan⁡(2π−θ)=−tan⁡θ\begin{align*} \sin\bigl(\tfrac{\pi}{2}-\theta\bigr) &= \cos\theta, & \cos\bigl(\tfrac{\pi}{2}-\theta\bigr) &= \sin\theta, & \tan\bigl(\tfrac{\pi}{2}-\theta\bigr) &= \cot\theta \\ \sin\bigl(\tfrac{\pi}{2}+\theta\bigr) &= \cos\theta, & \cos\bigl(\tfrac{\pi}{2}+\theta\bigr) &= -\sin\theta, & \tan\bigl(\tfrac{\pi}{2}+\theta\bigr) &= -\cot\theta \\ \sin(\pi-\theta) &= \sin\theta, & \cos(\pi-\theta) &= -\cos\theta, & \tan(\pi-\theta) &= -\tan\theta \\ \sin(\pi+\theta) &= -\sin\theta, & \cos(\pi+\theta) &= -\cos\theta, & \tan(\pi+\theta) &= \tan\theta \\ \sin\bigl(\tfrac{3\pi}{2}-\theta\bigr) &= -\cos\theta, & \cos\bigl(\tfrac{3\pi}{2}-\theta\bigr) &= -\sin\theta \\ \sin(2\pi-\theta) &= -\sin\theta, & \cos(2\pi-\theta) &= \cos\theta, & \tan(2\pi-\theta) &= -\tan\theta \end{align*}

Memory device:

  1. Odd multiples of π2\dfrac{\pi}{2}: ratios become co-ratios (sin⁡↔cos⁡\sin\leftrightarrow\cos, etc.).
  2. Even multiples of π2\dfrac{\pi}{2}: ratios stay the same type.
  3. Sign follows the quadrant of the terminal arm (ASTC: All–Sin–Tan–Cos positive).
Angle formQuadrant
π2−θ\dfrac{\pi}{2}-\thetaI
π2+θ\dfrac{\pi}{2}+\theta or π−θ\pi-\thetaII
π+θ\pi+\theta or 3π2−θ\dfrac{3\pi}{2}-\thetaIII
3π2+θ\dfrac{3\pi}{2}+\theta or 2π−θ2\pi-\thetaIV

Example 3

(i) sin⁡225∘=sin⁡(180∘+45∘)=−sin⁡45∘=−12\sin 225^\circ = \sin(180^\circ+45^\circ) = -\sin 45^\circ = -\dfrac{1}{\sqrt{2}}

(ii) tan⁡600∘=tan⁡60∘=3\tan 600^\circ = \tan 60^\circ = \sqrt{3}

(iii) cot⁡(−225∘)=−1\cot(-225^\circ) = -1

(iv) cosec⁡(−420∘)=−23\operatorname{cosec}(-420^\circ) = -\dfrac{2}{\sqrt{3}}

Example 4

sin⁡(180∘−θ)cos⁡(360∘−θ)tan⁡(90∘+θ)sin⁡(90∘−θ)cos⁡(180∘+θ)tan⁡(270∘−θ)=tan⁡θ\dfrac{\sin(180^\circ-\theta)\cos(360^\circ-\theta)\tan(90^\circ+\theta)}{\sin(90^\circ-\theta)\cos(180^\circ+\theta)\tan(270^\circ-\theta)} = \tan\theta

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