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Mathematics · Introduction to Trigonometry · NCERT Examples

Example 3

A complete, independently verified solution for NCERT Example 3.

Mathematics · Chapter 8 · NCERT Examples

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NCERT Example

Example 3

trigonometric ratios · squares

In right triangle ACB, AB = 29 units, BC = 21 units and angle ABC = θ. Find (i) cos²θ + sin²θ and (ii) cos²θ − sin²θ.

Right triangle ACB, right-angled at C, with BC 21 units, AB 29 units, and angle theta at B.
Figure 8-10 from the supplied NCERT chapter.

Solution

Board-exam working:

AC=AB2−BC2=292−212=20AC=\sqrt{AB^2-BC^2}=\sqrt{29^2-21^2}=20
sin⁡θ=2029,cos⁡θ=2129\sin\theta=\frac{20}{29},\qquad \cos\theta=\frac{21}{29}

(i)

cos⁡2θ+sin⁡2θ=(2129)2+(2029)2=841841=1\cos^2\theta+\sin^2\theta=\left(\frac{21}{29}\right)^2+\left(\frac{20}{29}\right)^2=\frac{841}{841}=1

(ii)

cos⁡2θ−sin⁡2θ=212−202292=41841\cos^2\theta-\sin^2\theta=\frac{21^2-20^2}{29^2}=\frac{41}{841}

Final answer

(i) 1

(ii) 41841\text{(ii) }\frac{41}{841}