# trigonometry equation

• Apr 15th 2007, 09:40 AM
sillyhacker
trigonometry equation
Hello,

I am a tad stuck on this question, so any help would be welcome.

y = (2x – 1)Tan 2x

Has a minimum Turing point at P, the x co-ordinate of P is K

Show that K satisfies the following

4K + sin 4K -2 = 0

thank you for the help

• Apr 15th 2007, 11:02 AM
topsquark
Quote:

Originally Posted by sillyhacker
Hello,

I am a tad stuck on this question, so any help would be welcome.

y = (2x – 1)Tan 2x

Has a minimum Turing point at P, the x co-ordinate of P is K

Show that K satisfies the following

4K + sin 4K -2 = 0

thank you for the help

Take the first derivative:
y = (2x - 1)*tan(2x)

y' = 2*tan(2x) + (2x - 1)* sec^2(2x)*2

At the minimum point y' = 0, so given the x = K:
2*tan(2K) + (2K - 1)* sec^2(2K)*2 = 0

Now, tan(2k) = sin(2K)/cos(2K) and sec(K) = 1/cos(K), so:
2*sin(2K)/cos(2K) + 2*(2K - 1)/cos^2(2K) = 0

Now multiply both sides by cos^2(2K):
2*sin(2K)*cos(2K) + 2*(2K - 1) = 0

But 2*sin(2K)*cos(2K) = sin(4K), so
sin(4K) + 4K - 2 = 0