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

1 Answer
Apr 7, 2017

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.