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Question

Question: The value \(\cos 105{^\circ} + \sin 105{^\circ}\)is...

The value cos⁡105∘+sin⁡105∘\cos 105{^\circ} + \sin 105{^\circ}is

A

12\frac{1}{2}

B

1

C

2\sqrt{2}

D

12\frac{1}{\sqrt{2}}

Answer

12\frac{1}{\sqrt{2}}

Explanation

Solution

cos⁡105∘+sin⁡105∘=cos⁡(90∘+15∘)+sin⁡(90∘+15∘)\cos 105{^\circ} + \sin 105{^\circ} = \cos(90{^\circ} + 15{^\circ}) + \sin(90{^\circ} + 15{^\circ})

=cos⁡15∘−sin⁡15∘=3+122−3−122=222=12\cos 15{^\circ} - \sin 15{^\circ} = \frac{\sqrt{3} + 1}{2\sqrt{2}} - \frac{\sqrt{3} - 1}{2\sqrt{2}} = \frac{2}{2\sqrt{2}} = \frac{1}{\sqrt{2}}.