How to use the half angle identity to find sin 105?

Question

Sydnee Glover · Accepted Answer

The formula for the half-angle must be applied:

\sin (\frac{θ}{2}) = \pm \sqrt{\frac{1 - \cos θ}{2}}

In this case, we want to find

\sin (105^{\circ})

, so that's what we want

\sin (\frac{θ}{2})

to equal. To find out what our

θ

is, set these to equal to each other:

\sin (105^{\circ}) = \sin (\frac{θ}{2})

105^{\circ} = \frac{θ}{2}

210^{\circ} = θ

This is our

θ

. Now, we can use the half angle formula:

= \sin (105^{\circ})

= \sin (\frac{210^{\circ}}{2})

= \pm \sqrt{\frac{1 - \cos (210^{\circ})}{2}}

= \pm \sqrt{\frac{1 - (- \frac{\sqrt{3}}{2})}{2}}

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

= \pm \sqrt{\frac{2 + \sqrt{3}}{4}}

= \pm \frac{\sqrt{2 + \sqrt{3}}}{\sqrt{4}}

= \pm \frac{\sqrt{2 + \sqrt{3}}}{2}

Since

105^{\circ}

is in quadrant II, we know that our answer will be positive that angle is above the

x

-axis (and we are taking the sine). Hence:

{\sin 105}^{\circ} = \frac{\sqrt{2 + \sqrt{3}}}{2}

Answered question