Evaluate the following: If (A+B)= + and (A-B)= + , Find the value… - Vidyadip

Question

Evaluate the following:
If $\sin(\text{A}+\text{B})=\sin\text{A}\cos\text{B}+\cos\text{A}\sin\text{B}$ and $\cos(\text{A}-\text{B})=\cos\text{A}\cos\text{B}+\sin\text{A}\sin\text{B},$ Find the values of:

$\sin75^\circ$
$\cos15^\circ.$

✓

Answer

Let $\text{A}=45^\circ$ and $\text{B}=30^\circ$
As, $\sin(\text{A}+\text{B})=\sin\text{A}\cos\text{B}+\cos\text{A}\sin\text{B}$
$\Rightarrow\sin(45^\circ+30^\circ)=\sin45^\circ\cos30^\circ+\cos45^\circ\sin30^\circ$
$\Rightarrow\sin(75^\circ)=\frac{1}{\sqrt{2}}\times\frac{\sqrt{3}}{2}+\frac{1}{\sqrt{2}}\times\frac12$
$\Rightarrow\sin75^\circ=\frac{\sqrt{3}}{2\sqrt{2}}+\frac{1}{2\sqrt{2}}$
$\therefore\ \sin75^\circ=\frac{\sqrt{3}+1}{2\sqrt{2}}$
As, $\cos(\text{A}-\text{B})=\cos\text{A}\cos\text{B}+\sin\text{A}\sin\text{B}$
$\Rightarrow\cos(45^\circ-30^\circ)=\cos45^\circ\cos30^\circ+\sin45^\circ\sin30^\circ$
$\Rightarrow\cos(15^\circ)=\frac{1}{\sqrt{2}}\times\frac{\sqrt{3}}{2}+\frac{1}{\sqrt{2}}\times\frac12$
$\Rightarrow\cos15^\circ=\frac{\sqrt{3}}{2\sqrt{2}}+\frac{1}{2\sqrt{2}}$
$\therefore\ \cos15^\circ=\frac{\sqrt{3}+1}{2\sqrt{2}}$
Disclaimer: $\cos15^\circ$ can also be calculated by taking A = 60° and B = 45°.

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