using exact values for the sine and cosine of both 3pi/4 and pi/3 and the angle difference identity for cosine, find the exact value of cos(5pi/12)

Question

also exact value of cos   95pi/6) and use this in the half angle identity for sine to find the exact value of sin(5pi/12)

Answer 1

Question 1

The angle difference identity for cosine is cos(x-y)=cos(x)cos(y)+sin(x)sin(y).

Then cos(5π/12) = cos(9π/12 - 4π/12)

= cos(3π/4 - π/3)

= cos(3π/4)cos(π/3)+sin(3π/4)sin(π/3)

= (-1/√2)(1/2)+(1/√2)(√3/2)

= (√3 - 1)/(2√2)

or you can rationalize the denominator to get the equivalent expression (√6 - √2)/4.

Question 2

The half angle identity for sine is sin(x/2)=±√((1-cos x)/2). Letting x=5π/6, we have x/2=5π/12. Note that 5π/12 is in the first quadrant of the unit circle (it's just a little smaller than π/2) so its sine is positive.

cos(5π/12) = cos((5π/6)/2) = +√((1-cos(5π/6))/2) = √((1-(-√3/2))/2) = √((1+√3/2)/2) = √((2+√3)/4) = (√(2+√3))/2.

This answer has a square root nested inside another square root. One way to get an answer without nested square roots is to the use the same method as Question 1 above, but use the angle difference identity for sine instead of cosine. Then you will get (√6 + √2)/4.

You can verity that (√(2+√3))/2 = (√6 + √2)/4 by squaring both sides.