# trigonometry derivative

• May 5th 2008, 12:42 AM
autkcat
trigonometry derivative
(Hi)
1. Use the inequalities sinx<x<tanx for a suitable value of x to show that pi lies between 3 and 2root3.

2.By writing tanx as sinx/cosx, show that d(lntanx)/dx = 2cosec2x.

3. A tuning fork sounding A avobe middle C oscillates 440 times a second. The displacement of the tip of the tuning fork is given by 0.02cos(2pi x 440t) millimetres, where t is the time in seconds after it is activated. Find
a. the greatest speed, b. the greatest acceleration of the tip as it is oscillates.

Thanks!
• May 5th 2008, 01:08 AM
Jhevon
Quote:

Originally Posted by autkcat
(Hi)
1. Use the inequalities sinx<x<tanx for a suitable value of x to show that pi lies between 3 and 2root3.

plug in x = pi/6
• May 5th 2008, 01:12 AM
Jhevon
Quote:

Originally Posted by autkcat
2.By writing tanx as sinx/cosx, show that d(lntanx)/dx = 2cosec2x.

what have you tried

follow the hint (though i don't think it helps--if you know the derivative of tan(x), it's easier to do it straight): $\displaystyle \frac d{dx} \ln \tan x = \frac d{dx} \ln \frac {\sin x}{\cos x} = \frac d{dx} ( \ln \sin x - \ln \cos x)$

now use the chain rule and simplify
• May 5th 2008, 03:20 AM
topsquark
Quote:

Originally Posted by autkcat
3. A tuning fork sounding A avobe middle C oscillates 440 times a second. The displacement of the tip of the tuning fork is given by 0.02cos(2pi x 440t) millimetres, where t is the time in seconds after it is activated. Find
a. the greatest speed, b. the greatest acceleration of the tip as it is oscillates.

Speed is v = dx/dt. So what is the largest value of dx/dt?

Acceleration is a = dv/dt. What is the greatest value of dv/dt?

-Dan
• May 6th 2008, 12:16 AM
autkcat
2.By writing tanx as sinx/cosx, show that d(lntanx)/dx = 2cosec2x.

I think i figured this out. the derivative of tanx is sec^2.
So, d(lntanx)/dx = sec^2x(cosx/sinx)
= 1/cosxsinx
= 1/(0.5sin2x) [double angle formula]
= 2cosec2x

Right??