Question

Find the exact value of the expression:

$\sin {105}^{\circ }$

Short answer

$\frac{1}{4}(\sqrt{6}+\sqrt{2})$

Step-by-step solution

Step 1. Given information 

We have been given an expression $\sin {105}^{\circ }$.

We have to find its exact value.

Step 2. Simplify the expression

We can also write the expression as:

$\sin {105}^{\circ }=\sin (135°-{30}^{°})$

Step 3.  Apply the compound angle formula

$\sin (135°-{30}^{°})$has the form role="math" $\sin (\alpha -\beta )$

Therefore, apply

$\sin \left(\alpha +\beta \right)=\sin \alpha \cos \beta -\cos \alpha \sin \beta$

localid="1646471429976" $\begin{array}{r}\sin {135}^{\circ }\cos {30}^{\circ }-\cos {135}^{\circ }\sin {30}^{\circ }=\left(\left(\frac{\sqrt{2}}{2}\right)\cdot \left(\frac{\sqrt{3}}{2}\right)\right)-\left(\left(-\frac{\sqrt{2}}{2}\right)\cdot \left(\frac{1}{2}\right)\right)\\ =\left(\frac{\sqrt{6}}{4}\right)-\left(-\frac{\sqrt{2}}{4}\right)\\ =\left(\frac{\sqrt{6}+\sqrt{2}}{4}\right)\\ =\frac{1}{4}(\sqrt{6}+\sqrt{2})\end{array}$