Evaluate sin 60° cos 30° + sin 30°cos 60°. What is the value of sin(60° + 30°). What can you conclude?

Let us first solve sin 60° cos 30° + sin 30° cos 60°.

We know,

sin 60° = √3/2

cos 30° = √3/2

sin 30° = 1/2

& cos 60° = 1/2

So, sin 60° cos 30° + sin 30° cos 60° =

⇒ sin 60° cos 30° + sin 30° cos 60° =

⇒ sin 60° cos 30° + sin 30° cos 60° = 1 …(i)

Now, for sin (60° + 30°):

sin (60° + 30°) = sin 90°

⇒ sin (60° + 30°) = 1 [∵, sin 90° = 1] …(ii)

By equations (i) & (ii), we can conclude that

sin (60° + 30°) = sin 60° cos 30° + sin 30° cos 60°

And infact in general, let 60° = x and 30° = y. Then,

sin (x + y) = sin x cos y + sin y cos x