Expert: Paul Klarreich - 1/2/2008

Question
Hello, Paul.
Can you solve (sin(x/2))^2 = 2sinx.
This looks really easy but I can't solve it!
I can't isolate a single trig function as the subject.
Please help!
Thanks.

Answer
Questioner:   Chris
Category:  Advanced Math
Private:  No
 
Subject:  Mathematics - Trigomometry.
Question:  Hello, Paul.
Can you solve (sin(x/2))^2 = 2sinx.
This looks really easy but I can't solve it!
I can't isolate a single trig function as the subject.
Please help!
Thanks.
..................................
Hi, Chris,

It's not so easy, but here is a formula that you will need later on -- one of the 'half-angle' rules:

                1 - cos x
sin(x/2) = sqrt(------------)
                   2

                1 - cos x
[sin(x/2)]^2 =  -----------
                   2
Now that is the left side of your equation:(turned around.)

           1 - cos x
2 sin x =  -----------
               2
[Using C as an abbreviation.]

                 1 - C
2 sqrt(1 - C^2) = -------
                   2
            1 - 2C + C^2
4(1 - C^2) = -------------
                  4

16 - 16C^2 = 1 - 2C + C^2

0 = -15 - 2C + 17C^2

17C^2 - 2C - 15 = 0

(17C + 15)(C - 1) = 0

C = -15/17

C = 1

I'll leave the rest to you.