Prove value of $\sin(15°)$

Question

I'm doing the following exercise: prove that $$ \sin(15°)=\frac{1}{2\sqrt{2+\sqrt{3}}} $$

I'm using this formula: $$ \sin(a-b)=\sin(a)\cos(b)-\cos(a)\sin(b) $$

to get: $$ \sin(45-30)=\sin(45)\cos(30)-\cos(45)\sin(30) \\ =\frac{1}{\sqrt2}\frac{\sqrt3}{2}-\frac{1}{\sqrt2}\frac{1}{2} \\ =\frac{\sqrt3-1}{2\sqrt2} $$

However I'm stuck, can't seem to get to the desired result. I tried multiplying for $$\frac{\sqrt{3}+1}{\sqrt{3}+1}$$ but didn't get anything out of it. Any help will be really appreciated.

Answer 1

They are equivalent. Since $$(1 + \sqrt{3})^2 = 1 + 2 \sqrt{3} + 3 = 2(2 + \sqrt{3}),$$ it follows that $$\sqrt{2 + \sqrt{3}} = \frac{1 + \sqrt{3}}{\sqrt{2}}.$$ Then $$\frac{1}{2 \sqrt{2 + \sqrt{3}}} = \frac{\sqrt{2}}{2 (1 + \sqrt{3})} = \frac{\sqrt{3} - 1}{2 \sqrt{2}}.$$

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It is also worth noting that we can obtain the desired expression more directly via the half-angle identity for $0 \le \theta \le 90^\circ$: $$\sin \frac{\theta}{2} = \sqrt{\frac{1 - \cos \theta}{2}} = \frac{\sqrt{1 - \cos^2 \theta}}{\sqrt{2 (1 + \cos \theta)}} = \frac{\sin \theta}{\sqrt{2 (1 + \cos \theta)}},$$ where upon selecting $\theta = 30^\circ$ immediately yields $$\sin 15^\circ = \frac{1}{2 \sqrt{2 + \sqrt{3}}}$$

Answer 2

Since, ${\sqrt3 - 1} = \sqrt{(\sqrt3 - 1)^2}= \sqrt2 \sqrt{2-\sqrt3}$ $$\frac{\sqrt3 - 1}{2\sqrt2} = \frac{\sqrt{2 - \sqrt3}}{2}$$ Rationalising the numerator $$\frac{\sqrt{2 - \sqrt3}}{2}\cdot \frac{\sqrt{2+\sqrt3}}{\sqrt{2+\sqrt3}}= \frac{1}{2\sqrt{2+\sqrt3}}$$

Answer 3

Because $$\frac{\sqrt3-1}{2\sqrt2}=\frac{\sqrt3-1}{\sqrt2(\sqrt3+1)(\sqrt3-1)}=\frac{1}{\sqrt2(\sqrt3+1)}=$$ $$=\frac{1}{\sqrt2\sqrt{(\sqrt3+1)^2}}=\frac{1}{\sqrt2\cdot\sqrt{4+2\sqrt3}}=\frac{1}{2\sqrt{2+\sqrt3}}.$$

Answer 4

Let's start from where you stack; after rationalization of denominator by $\sqrt 3 +1$, you got:

$$sin (15)=\frac{\sqrt 2}{2(\sqrt 3 +1)}=\frac{1}{2\times \frac{\sqrt 3 +1}{\sqrt 2}}$$

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