Mathematics · Introduction to Trigonometry · NCERT Examples

Example 9

A complete, independently verified solution for NCERT Example 9.

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Mathematics · Chapter 8 · NCERT Examples

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NCERT Example

Example 9

trigonometric identities · in terms of sin

Express cos A, tan A and sec A in terms of sin A.

Expressing ratios through sin AA right triangle marks angle A, its opposite side, adjacent side and hypotenuse so the trigonometric ratios can be read directly.adjacentoppositehypotenuseAABC
Expressing ratios through sin A — explanatory sketch (not to scale).

Solution

Board-exam working:

sin⁡2A+cos⁡2A=1\sin^2A+\cos^2A=1
cos⁡2A=1−sin⁡2A\cos^2A=1-\sin^2A

Since A is acute, cos A is positive.

cos⁡A=1−sin⁡2A\cos A=\sqrt{1-\sin^2A}
tan⁡A=sin⁡Acos⁡A=sin⁡A1−sin⁡2A\tan A=\frac{\sin A}{\cos A}=\frac{\sin A}{\sqrt{1-\sin^2A}}
sec⁡A=1cos⁡A=11−sin⁡2A\sec A=\frac1{\cos A}=\frac1{\sqrt{1-\sin^2A}}

Final answer

cos⁡A=1−sin⁡2A,tan⁡A=sin⁡A1−sin⁡2A,sec⁡A=11−sin⁡2A\cos A=\sqrt{1-\sin^2A},\quad \tan A=\frac{\sin A}{\sqrt{1-\sin^2A}},\quad \sec A=\frac1{\sqrt{1-\sin^2A}}

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