Expert: Ahmed Salami - 4/17/2009

Question
Hey I have some questions to find the exact values, I'm not sure about the steps involved. Could you please show me step by step... Thanks heaps...

☻ sin 22.5 cos 22.5
☻ cos^2 15 - sin^2 15
☻ 2 cos^2 22.5 -1
☻ 1- 2sin^2 75


Answer
Hi Sarah,
You have to make use of the double angle formulas here.
sin(A + B) = sinA.cosB + sinB.cosA
sin(A - B) = sinA.cosB - sinB.cosA
cos(A + B) = cosA.cosB - sinA.sinB
cos(A - B) = cosA.cosB + sinA.sinB
as well as the identity
cos^2 A + sin^2 A = 1
Now, lets take A = B and make use of the first and third formulas, we have
sin(A + A) = sinA.cosA + sinA.cosA
sin2A = 2sinA.cosA                                       (i)
and
cos(A + A) = cosA.cosA - sinA.sinA
cos2A = cos^2 A - sin^2 A                                (ii)
     = cos^2 A - (1 - cos^2 A)
     = cos^2 A - 1 + cos^2 A
     = 2cos^2 A - 1                                     (iii)
     = 2(1 - sin^2 A) - 1
     = 2 - 2sin^2 A - 1
     = 1 - 2sin^2 A                                     (iv)
Notice that cos^2 A can be written in three different ways. These formulas are common and can be found in all trigonometric texts.
Anyway, from here on you can find your exact values.
1)Using (i)
2.sin 22.5 cos 22.5 = sin[2(22.5)]
sin 22.5 cos 22.5 = (1/2)sin[2(22.5)]
                 = (1/2)sin(45)
                 = (1/2)(sqrt2)/2
                 = (sqrt2)/4

2)Using (ii)
cos^2 15 - sin^2 15 = cos[2(15)]
                   = cos(30)
                   = (sqrt3)/2

3)Using (iii)
2cos^2 (22.5) - 1 = cos[2(22.5)]
                 = cos(45)
                 = (sqrt2)/2

4)Using (iv)
1 - 2sin^2 (75) = cos[2(75)]
               = cos(150)
               = -(sqrt3)/2    [i.e -cos(180-150) = -cos(30)]

Hope it helps.

Regards