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  • Please solve RD Sharma class 12 chapter 26 Direction Cosines and Direction Ratios exercise Fill in the blanks question 5 maths textbook solution

Please solve RD Sharma class 12 chapter 26 Direction Cosines and Direction Ratios exercise Fill in the blanks question 5 maths textbook solution

Answers (1)

Final Answer : 2

Hint:

l^{2}+m^{2}+n^{2}=1

Given:

Line makes angles α,β, γ with x,y,z axes

To Find: 

\sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma

Solution:

Direction cosine of a line making angles α,β,y are

\begin{aligned} &l=\cos \alpha \\ &m=\cos \beta \text { and } \end{aligned}

\begin{aligned} &n=\cos \gamma \text { and } \\ &l^{2}+m^{2}+n^{2}=1 \\ &\therefore \cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1 \end{aligned}        ......(i)

Now as\left[\cos ^{2} x=1-\sin ^{2} x\right]                ....... (ii)

∴ Using above eq(ii) in (i), we get

\begin{aligned} &\left(1-\sin ^{2} \alpha\right)+\left(1-\sin ^{2} \beta\right)+\left(1-\sin ^{2} x\right)=1 \\\\ &\Rightarrow 3-\left(\sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma\right)=1 \\\\ &\Rightarrow \sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma=3-1 \\\\ &\Rightarrow \sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma=2 \end{aligned}

Therefore, the value of  \sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma \text { is } 2.

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