The direction angles made by a line ‘ L ’ with the coordinate axes, are α,β,γ.  Then the range of sin2α+sin2β

# The direction angles made by a line ' $L$ ’ with the coordinate axes, are $\alpha ,\beta ,\gamma$.  Then the range of ${\mathrm{sin}}^{2}\alpha +{\mathrm{sin}}^{2}\beta$

1. A

$\varphi$

2. B

$\left(0,1\right)$

3. C

$\left[1,2\right]$

4. D

$ℝ$

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### Solution:

If$\alpha ,\beta ,\gamma$ are the angles made by a line ‘L’ with the coordinate axes in the positive direction then${\mathrm{cos}}^{2}\alpha +{\mathrm{cos}}^{2}\beta +{\mathrm{cos}}^{2}\gamma =1$

It implies that $1-{\mathrm{sin}}^{2}\alpha +1-{\mathrm{sin}}^{2}\beta +1-{\mathrm{sin}}^{2}\gamma =1⇒{\mathrm{sin}}^{2}\alpha +{\mathrm{sin}}^{2}\beta +{\mathrm{sin}}^{2}\gamma =2$

Hence, ${\mathrm{sin}}^{2}\alpha +{\mathrm{sin}}^{2}\beta =2-{\mathrm{sin}}^{2}\gamma$

The minimum value of ${\mathrm{sin}}^{2}\gamma$is zero and its maximum value is 1

So that the range of  ${\mathrm{sin}}^{2}\alpha +{\mathrm{sin}}^{2}\beta$is $\overline{)\left[1,2\right]}$  +91

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