Question

How do you find the exact values of the sine, cosine, and tangent of the angle `15^circ`?

Answer 1

Working below

Explanation:

FACTS TO KNOW:
`sin(A-B) = sinAcosB-cosAsinB`
`cos(A-B) = cosAcosB+sinAsinB`
`tanx=sinx/cosx`
`sin45=cos45=1/sqrt2`
`sin30=cos60=1/2`
`sin60=cos30=sqrt3/2`

To find `sin15`, we evaluate `sin(45-30)`:
`sin(15)=sin(45-30)`
`=sin45cos30-cos45sin30`
`=1/sqrt2*sqrt3/2-1/sqrt2*1/2`
`=(sqrt3-1)/(2sqrt2)`

To find `cos15`, we evaluate `cos(45-30)`:
`cos(15)=cos(45-30)`
`=cos45cos30+sin45sin30`
`=1/sqrt2*sqrt3/2+1/sqrt2*1/2`
`=(sqrt3+1)/(2sqrt2)`

To find `tan15`, we evaluate `sin15/cos15`
`tan15=sin15/cos15`
`=((sqrt3-1)/(2sqrt2))/((sqrt3+1)/(2sqrt2))`
`=(sqrt3-1)/(sqrt3+1)`

Note that it is possible to simplify further by rationalising the denominator of each.