Here is an example of using a sum identity:

Find #sin15^@#.

If we can find (think of) two angles #A# and #B# whose sum or whose difference is 15, and whose sine and cosine we know.

#sin(A-B)=sinAcosB-cosAsinB#

We might notice that #75-60=15#

so #sin15^@=sin(75^@-60^@)=sin75^@cos60^@-cos75^@sin60^@#

BUT we don't know sine and cosine of #75^@#. So this won't get us the answer. (I included it because when solving problems we DO sometimes think of approaches that won't work. And that's OK.)

#45-30=15# and I do know the trig functions for #45^@# and #30^@#

#sin15^@=sin(45^@-30^@)=sin45^@cos30^@-cos45^@sin30^@#

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

#=(sqrt6 - sqrt 2)/4#

There are other way of writing the answer.

**Note 1**

We could use the same two angles and the identity for #cos(A-B)# to find #cos 15^@#

**Note 2**

Instead of #45-30=15# we could have used #60-45=15#

**Note 3**

Now that we have #sin 15^@# we could use #60+15=75# and #sin(A+B)# to find #sin75^@#. Although if the question had been to find #sin75^@, I'd probably use #30^@# and #45^@#