Question

How do you use the sum or difference identities to find the exact value of `sin165^circ`?

Answer 1

`color(blue)(sin165^circ=sin(180^circ-15^circ)=sin15^circ=(sqrt6-sqrt2)/4`

Explanation:

We know that ,

`color(red)((1)sin(x+y)=sinxcosy+cosxsiny`

Here ,

`sin165^circ=color(red)(sin(135^circ+30^circ)tocolor(red)(Apply(1)`

`sin165^circ=color(red)(sin135^circcos30^circ+cos135^circsin30^circ`

`sin165^circ=1/sqrt2*sqrt3/2+(-1/sqrt2)*1/2`

`sin165^circ=1/sqrt2*sqrt3/2-1/sqrt2*1/2`

`sin165^circ=(sqrt3-1)/(2sqrt2)`

`sin165^circ=(sqrt3-1)/(2sqrt2)xxsqrt2/sqrt2`

`sin165^circ=(sqrt6-sqrt2)/4`

OR

`color(blue)(sin165^circ=sin(180^circ-15^circ)=sin15^circ=(sqrt6-sqrt2)/4`