Maths Trig identities

Question

Exact value of Sin pi/12

 

I got this answer to be  (√6 - √2 )/4 (part a)

 

I then done the half angled identity to be  √(2 - √3)/2 (part b)

 

Both these expressions are equivalent and I need to prove it so the question is 

 

By squaring the exact values obtained for sin(π/12) in parts (a)
and (b) (or otherwise), show that the two expressions are equivalent.

 

Really don't understand this any help would be great.

Answer 1

If you use the half-angle identity, you'll be evaluating at the well-known angle pi/6.

 

sin([pi/6]/2) = √(1 - cos(pi/6))/2)

 

= √(1 - √(3)/2)/2)

 

= √(1/2 - √(3)/4)

 

Squaring this result yields: 1/2 - √(3)/4

 

If you use the sin(u - v) formula, you'll be evaluating sin(pi/3 - pi/4)

 

sin(pi/3 - pi/4) = sin(pi/3)cos(pi/4) - sin(pi/4)cos(pi/3)

 

= (√(3)/2)*(√(2)/2) - (√(2)/2)*(1/2)

 

= √(6)/4 - √(2)/4

 

= 1/4*(√(6) - √(2))

 

Squaring this result yields something a little more tricky. Gonna have to do some FOIL'ing

 

1/4*1/4*(√(6) - √(2))*(√(6) - √(2))

 

= 1/16*(6 - √(12) - √(12) + 2)

 

= 1/16*(8 - 2√(12))

 

= 1/16*(8 - 2*2*√(3))

 

= 8/16 - 4*√(3)/16

 

= 1/2 - √(3)/4

 

This matches your first result, thus proving it works. Hope this helps.