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  • Explain solution RD Sharma class 12 chapter 26 Direction Cosines and Direction Ratios exercise Fill in the blanks question 4 maths

Explain solution RD Sharma class 12 chapter 26 Direction Cosines and Direction Ratios exercise Fill in the blanks question 4 maths

Answers (1)

Final Answer :  -1

Hint:

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

Given:

Line makes angles \alpha, \beta, \gamma \text { with } x, y, z \text { axes }

To Find: 

\cos 2 \alpha+\cos 2 \beta+\cos 2 \gamma

Solution:

Direction cosine of a line making angles \alpha ,\beta ,\gamma with co-ordinate axes are

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

\begin{aligned} &\text { and } l^{2}+m^{2}+n^{2}=1 \\ &\Rightarrow \cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1 \\ &\Rightarrow 2 \cos ^{2} \alpha+2 \cos ^{2} \beta+2 \cos ^{2} \gamma=2 \end{aligned}

Now, using the formula:  \left[1+\cos 2 x=2 \cos ^{2} x\right]

\begin{aligned} &\Rightarrow 1+\cos 2 \alpha+1+\cos 2 \beta+1+\cos 2 \gamma=2 \\ &\Rightarrow \cos 2 \alpha+\cos 2 \beta+\cos 2 y=2-3 \\ &\Rightarrow \cos 2 \alpha+\cos 2 \beta+\cos 2 y=-1 \end{aligned}

Therefore, value of (\cos 2 \alpha+\cos 2 \beta+\cos 2 y) \text { is }-1

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