Question

If sin (A − B) = sin A cos B − cos A sin B and cos (A − B) = cos A cos B + sin A sin B, find the values of sin 15° and cos 15°.

Answer 1

Given:

sin (A − B) = sin A cos B − cos A sin B   ......(1)

cos (A − B) = cos A cos B + sin A sin B ......(2)

`To find:

The values of `sin 15^@` and `cos 15^@`

In this problem, we need to find `sin 15^@` and `cos 15^@`

Hence to get `15^@` angle we need to choose the value if A and B such that `(A - B) = 15^@`

So If we choose  A = 45° and B = 30°

Then we get (A - B) = 15°

Therefore by substituting A = 45° and B = 30° in equation (1)

We get

`sin(45^@ - 30^@) = sin 45^@ cos 30^@ - cos 45^@ sin 30^@`

Therefore

`sin(15^@) = sin 45^@ cos 30^@ - cos 45^@ sin 30^@`  ....(3)

Now we know that,

`sin 45^@ = cos 45^@ = 1/sqrt2, sin 30^@ = 1/2, cos 30^@ = sqrt3/2`

Now by substituting above values in equation (3)

We get,

`sin (15^@) = (1/sqrt2) xx (sqrt3/2) - (1/sqrt2) xx (1/2)`

`= sqrt3/(2sqrt2) - 1/(2sqrt2)`

`= (sqrt3 - 1)/(2sqrt2)`

Therefore

`cos(15^@) = (sqrt3 -1)/(2sqrt2)`  ....(6)

Therefore from equation (4) and (6)

`sin(15^@) = (sqrt3 - 1)/(2sqrt2)`

`cos(15^@) = (sqrt3 + 1)/(2sqrt2)`