Question

Find the value of the following expression:

sin 45° cos 30° – cos 45° sin 30°

Answer 1

Given: sin 45° cos 30° – cos 45° sin 30°

Step-wise calculation:

1. Use the sine difference identity:

sin (A – B) = sin A cos B – cos A sin B

So, the expression equals sin (45° – 30°) = sin 15°.

2. Compute sin 15° by substituting standard values:

`sin 45^circ = sqrt(2)/2` 

`cos 30^circ = sqrt(3)/2`

`cos 45^circ = sqrt(2)/2` 

`sin 30^circ = 1/2` 

So, sin 45° cos 30° – cos 45° sin 30°

= `(sqrt(2)/2) (sqrt(3)/2) - (sqrt(2)/2) (1/2)`

= `(sqrt(2)/2) xx ((sqrt(3) - 1)/2)` 

= `(sqrt(2) (sqrt(3) - 1))/4` 

= `((sqrt(6) - sqrt(2)))/4`