How do you find the exact value of sin 105 degrees?

Question

How do you find the exact value of

\sin 105

degrees?

Answer 1

Use

\sin 105 = \sin (60 + 45) = \sin 60 \cos 45 + \cos 60 \sin 45

= (\frac{\sqrt{3}}{2}) (\frac{1}{\sqrt{2}}) + (\frac{1}{2}) (\frac{1}{\sqrt{2}}) = \frac{\sqrt{2}}{4} ((\sqrt{3} + 1) = 0.9656

nearly.
Explanation:

\sin 45, \cos 45

and

\sin 60

are irrational.

Answer 2

We know

\sin (A + B) = \sin A \cos B + \cos A \sin B

Hence

\sin 105^{\circ}

= \sin (60^{\circ} + 45^{\circ})

= \sin 60^{\circ} \cos 45^{\circ} + \cos 60^{\circ} \sin 45^{\circ}

= \frac{\sqrt{3}}{2} \times \frac{1}{\sqrt{2}} + \frac{1}{2} \times \frac{1}{\sqrt{2}}

= \frac{\sqrt{3} + 1}{2 \sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}

= \frac{\sqrt{6} + \sqrt{2}}{4}

Answers (2)