Evaluate sin 36° cos 9° + cos 36° sin 9°

Question

Evaluate:

\[ \sin 36^\circ \cos 9^\circ+\cos 36^\circ \sin 9^\circ \]

Solution

Using the identity:

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

Here,

\[ A=36^\circ,\qquad B=9^\circ \]

Therefore,

\[ \sin 36^\circ \cos 9^\circ+\cos 36^\circ \sin 9^\circ \]

\[ =\sin(36^\circ+9^\circ) \]

\[ =\sin 45^\circ \]

We know that:

\[ \sin 45^\circ=\frac{1}{\sqrt{2}} \]

Therefore,

\[ \boxed{\sin 36^\circ \cos 9^\circ+\cos 36^\circ \sin 9^\circ=\frac{1}{\sqrt{2}}} \]