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Some Application of Trigonomertry — Maths STD 10 — Question

CBSE BoardEnglish MediumSTD 10MathsSome Application of Trigonomertry1 Mark
Question
Write ‘True’ or ‘False’ and justify your answer.
If a man standing on a platform 3 metres above the surface of a lake observes a cloud and its reflection in the lake, then the angle of elevation of the cloud is equal to the angle of depression of its reflection.
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Answer

False. The observer is at the platform (P) 3m above the surface LK of the lake. He observes the angle of elevation of cloud C from P and its reflection image in the lake is formed at I. The observer measures in the angle of depression of image (I) $\theta_2.$ Draw PM $\perp$ on the vertical line passing through the cloud and its image.
CK = KI = x by the prop. of reflection. CM = CK - MK = x - 3 MI = KI + MK = x + 3 Now, $\tan\theta_1=\frac{\text{x}-3}{\text{y}}\text{ and }\tan\theta_2=\frac{\text{x}+3}{\text{y}}$ $\Rightarrow\ \text{y}=\frac{\text{x}-3}{\tan\theta_1}\text{ and }\text{y}=\frac{\text{x}+3}{\tan\theta_2}$ $\Rightarrow\ \frac{\text{x}+3}{\tan\theta_2}=\frac{\text{x}-3}{\tan\theta_1}$ $\Rightarrow\ \tan\theta_2=\Big(\frac{\text{x}+3}{\text{x}-3}\Big)\tan\theta_1$ $\Rightarrow\ \tan\theta_1\neq\tan\theta_2$or $\theta_1\neq\theta_2$
Alternate Answer
By the property of image formation, the distance of image and the object are equal from the reflecting surface. So, $\text{KC}=\text{KI}$ $\Rightarrow\ \text{MI}\neq\text{MC}$ $\Rightarrow\ \triangle\text{MPC}\neq\triangle\text{MPI}$ So $\theta_1\neq\theta_2$

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