Question

Determine o valor de sen 105 + cos 105

Answer 1

$\sin(105^\circ) = \sin(60^\circ + 45^\circ) = \\\\ \sin(60^\circ) \times \cos(45^\circ) + \cos(60^\circ) \times \sin(45^\circ) \\\\ = \frac{ \sqrt{3} }{2} \times \frac{ \sqrt{2} }{2} + \frac{1}{2} \times \frac{ \sqrt{2} }{2} \\ \\ \frac{ \sqrt{6}}{4} + \frac{ \sqrt{2} }{4} \\ \\ \boxed{\sin(105^\circ) = \frac{ \sqrt{6} + \sqrt{2} }{4}}$

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$\cos(105^\circ) = \cos(60^\circ + 45^\circ) = \\\\ \cos(60^\circ) \times \cos(45^\circ) - \sin(60^\circ) \times \sin(45^\circ) \\\\ \frac{1}{2} \times \frac{ \sqrt{2} }{2} - \frac{ \sqrt{3} }{2} \times \frac{ \sqrt{2} }{2} \\ \\ \frac{ \sqrt{2} }{4 } - \frac{ \sqrt{6} }{4} \\ \\ \boxed{\cos(105^\circ) = \frac{ \sqrt{2} - \sqrt{6} }{4}}$

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$\sin(105^\circ) + \cos(105^\circ) = \\\\ \frac{\sqrt{6}+\sqrt{2}}{4} + \frac{\sqrt{2}-\sqrt{6}}{4} = \\\\ \frac{\sqrt{6}+\sqrt{2} + \sqrt{2} - \sqrt{6}}{4} = \frac{2\sqrt{2}}{4} \\\\ \boxed{\sin(105^\circ) + \cos(105^\circ) = \frac{\sqrt{2}}{2}}$

Answer 2

Resposta:

$\sqrt 2 /2$

Explicação: