Using the exact values of the trigonometric ratios of pi/4 and pi/6,?

Question

find the exact values of sin5pi/12 and cospi/12.

help needed..confused :(

Answer 1

sin(5π/12)

= sin(2π/12 + 3π/12)

= sin(π/6 + π/4)

= sin(π/6)cos(π/4) + cos(π/6)sin(π/4)

= (1/2)(1/√2) + (√3/2)(1/√2)

= √2/4 + √6/4

= (√2+√6)/4

You can work out the other trigonometric ratios for 5π/12 in the same way

For the trig ratios of π/12, note that:

π/12

= 3π/12 - 2π/12

= π/4 - π/6

You can now use the trig formulae to calculate sin(π/12), cos(π/12) and tan(π/12) (and sec(π/12), csc(π/12) and cot(π/12) if you need to).

cos(π/12)

= cos(π/4 - π/6)

= cos(π/4)cos(π/6) + sin(π/4)sin(π/6)

= (1/√2)[1/2 + √3/2]

= (√2+√6)/4

Answer 2

sin(pi/3 + pi/12) =sin(pi/3)cos(pi/12) +sin(pi/12)cos(pi/3)

                        =sin(60)cos(15)+ sin(15)cos(60)

                        = (3)^1/2/2*(2+(3)^1/2)^1/2/2 +1/2*(2-(3)^1/2/2

                        

cos(pi/12) = cos(15) = (2+(3)^1/2)^1/2/2