cot 70^{circ} + 4 cos 70^{circ} ની કિંમત શોધો | vedclass

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Question

(B) ધારો કે $x = \cot 70^{\circ} + 4 \cos 70^{\circ}$.આને $x = \frac{\cos 70^{\circ}}{\sin 70^{\circ}} + 4 \cos 70^{\circ} = \frac{\cos 70^{\circ} + 4

Vedclass · Accepted Answer

$\sqrt{3}$

Vedclass · Answer

(B) ધારો કે $x = \cot 70^{\circ} + 4 \cos 70^{\circ}$.આને $x = \frac{\cos 70^{\circ}}{\sin 70^{\circ}} + 4 \cos 70^{\circ} = \frac{\cos 70^{\circ} + 4 \sin 70^{\circ} \cos 70^{\circ}}{\sin 70^{\circ}}$ તરીકે લખી શકાય.$2 \sin 	heta \cos 	heta = \sin 2	heta$ નિત્યસમનો ઉપયોગ કરતા,$x = \frac{\cos 70^{\circ} + 2 \sin 140^{\circ}}{\sin 70^{\circ}}$.$\sin 140^{\circ} = \sin(180^{\circ} - 40^{\circ}) = \sin 40^{\circ}$ હોવાથી,$x = \frac{\cos 70^{\circ} + 2 \sin 40^{\circ}}{\sin 70^{\circ}}$.$\cos 70^{\circ} = \sin 20^{\circ}$ નો ઉપયોગ કરતા,$x = \frac{\sin 20^{\circ} + 2 \sin 40^{\circ}}{\sin 70^{\circ}} = \frac{\sin 20^{\circ} + \sin 40^{\circ} + \sin 40^{\circ}}{\sin 70^{\circ}}$.$\sin C + \sin D = 2 \sin \frac{C+D}{2} \cos \frac{C-D}{2}$ નો ઉપયોગ કરતા,$\sin 20^{\circ} + \sin 40^{\circ} = 2 \sin 30^{\circ} \cos 10^{\circ} = 2 	imes \frac{1}{2} 	imes \cos 10^{\circ} = \cos 10^{\circ}$.તેથી,$x = \frac{\cos 10^{\circ} + \sin 40^{\circ}}{\sin 70^{\circ}} = \frac{\sin 80^{\circ} + \sin 40^{\circ}}{\sin 70^{\circ}}$.$\sin C + \sin D = 2 \sin \frac{C+D}{2} \cos \frac{C-D}{2}$ નો ઉપયોગ કરતા,$x = \frac{2 \sin 60^{\circ} \cos 20^{\circ}}{\sin 70^{\circ}} = \frac{2 	imes \frac{\sqrt{3}}{2} 	imes \cos 20^{\circ}}{\cos 20^{\circ}} = \sqrt{3}$.

$\cot 70^{\circ} + 4 \cos 70^{\circ}$ ની કિંમત શોધો.

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