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  • Explain solution RD Sharma class 12 chapter 26 Direction Cosines and Direction Ratios exercise Fill in the blanks question 16 maths

Explain solution RD Sharma class 12 chapter 26 Direction Cosines and Direction Ratios exercise Fill in the blanks question 16 maths

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Final Answer:  \frac{\pi }{2}

Hint:

Use property \left[\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1\right]

Given:

\alpha+\beta=\frac{\pi}{2}

To Find: 

\gamma  (Angle between line and z axis)

Solution:

If a line makes α,β,γ, then

\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1            ......(i)

It is given that    \alpha+\beta=\frac{\pi}{2}

\alpha=\frac{\pi}{2}-\beta

Taking cosine both side

\begin{aligned} &\cos \alpha=\cos \left(\frac{\pi}{2}-\beta\right) \\\\ &\cos \alpha=\sin \beta \\\\ &\cos ^{2} \alpha=\sin ^{2} \beta \end{aligned}                        \left[\sin ^{2} x+\cos ^{2} x=1\right]

\begin{aligned} &\cos ^{2} \alpha=1-\cos ^{2} \beta \\\\ &\cos ^{2} \alpha+\cos ^{2} \beta=1 \end{aligned}                .......(ii)

Using the above equation (ii) in (i) we get

\begin{aligned} &\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1 \\\\ &1+\cos ^{2} x=1 \\\\ &\cos ^{2} x=0 \end{aligned}

\gamma=\frac{\pi}{2}\\\\ or \; \; 90^{\circ}
Therefore, the value of \gamma is \frac{\pi }{2}

 

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