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Chapter 8 · Question 4

State the trigonometric ratios of complementary angles.

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Answer

Direct Answer

For acute AA, sin⁡(90∘−A)=cos⁡A\sin(90^\circ-A)=\cos A, cos⁡(90∘−A)=sin⁡A\cos(90^\circ-A)=\sin A, tan⁡(90∘−A)=cot⁡A\tan(90^\circ-A)=\cot A, cot⁡(90∘−A)=tan⁡A\cot(90^\circ-A)=\tan A, sec⁡(90∘−A)=cosec⁡A\sec(90^\circ-A)=\cosec A, and cosec⁡(90∘−A)=sec⁡A\cosec(90^\circ-A)=\sec A.

Simple Explanation

Ratios swap for complementary angles: sine with cosine, tangent with cotangent, secant with cosecant.

Exam-Ready Structure

In a right triangle, the two acute angles are complementary. The side opposite one acute angle is adjacent to the other. Therefore the ratios interchange: sine becomes cosine, tangent becomes cotangent, and secant becomes cosecant.

Key Points

  • For acute AA, sin⁡(90∘−A)=cos⁡A\sin(90^\circ-A)=\cos A, cos⁡(90∘−A)=sin⁡A\cos(90^\circ-A)=\sin A, tan⁡(90∘−A)=cot⁡A\tan(90^\circ-A)=\cot A, cot⁡(90∘−A)=tan⁡A\cot(90^\circ-A)=\tan A, sec⁡(90∘−A)=cosec⁡A\sec(90^\circ-A)=\cosec A, and cosec⁡(90∘−A)=sec⁡A\cosec(90^\circ-A)=\sec A.

Related Questions

Q5

Prove the identity sin⁡2A+cos⁡2A=1\sin^2 A+\cos^2 A=1.

Q6

Evaluate sin⁡230∘+cos⁡230∘\sin^2 30^\circ+\cos^2 30^\circ.

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