Introduction To Trigonometry Question - CBSE_NCERT_TEXTBOOK

Question

Consider  ACB, right-angled at C, in which AB = 29 units, BC = 21 units and  ABC =  (see Fig. 8.10). Determine the values of (i) cos2  + sin2 , (ii) cos2  – sin2 

Accepted Answer

1. Find the third side AC using Pythagoras:
$$AB^2 = AC^2 + BC^2 \Rightarrow AC^2 = 29^2 - 21^2 = 841 - 441 = 400$$
$$ herefore\; AC = \sqrt{400}=20 ext{ units}$$

2. Express sinθ and cosθ (θ is at vertex B):
- Opposite side to θ = AC = 20
- Adjacent side to θ = BC = 21
- Hypotenuse = AB = 29
$$\sin heta = \frac{ ext{opposite}}{ ext{hypotenuse}} = \frac{20}{29}$$
$$\cos heta = \frac{ ext{adjacent}}{ ext{hypotenuse}} = \frac{21}{29}$$

3. Compute $\cos^2 heta + \sin^2 heta$:
$$\cos^2 heta + \sin^2 heta = \left(\frac{21}{29} ight)^2 + \left(\frac{20}{29} ight)^2
= \frac{441 + 400}{29^2}
= \frac{841}{841}
= 1$$

4. Compute $\cos^2 heta - \sin^2 heta$:
$$\cos^2 heta - \sin^2 heta = \left(\frac{21}{29} ight)^2 - \left(\frac{20}{29} ight)^2
= \frac{441 - 400}{29^2}
= \frac{41}{841}$$
(This is also $\cos2 heta$.)

Thus the required values are obtained.

Question:

Step-by-Step Solution

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