Question

Assuming A≠0, evaluate the limit
lim 3sin^2(At) / cos(At) - 1
t-->0

Answer 1

Start with:
\[\sin^2(\theta) + \cos^2(\theta) = 1\]
Solve that for sin^2(theta), and substitute

Answer 2

use 1 - cos At = 2sin^2(At/2 )
also sin^2(At) = 4sin^2(At/2)*cos^2(At/2

Answer 3

lim 3sin^2(At) / cos(At) - 1
t-->0

= lim - 3 *4sin^2(At/2)*cos^2(At/2/ 2sin^2(At/2 )
t-->0
now you can do further

Answer 4

Alternate way:
\[3\frac{\sin^2(At)}{\cos(At) - 1} = 3\frac{\cos^2(At) - 1}{\cos(At) - 1} = 3\frac{(\cos(At)-1)(\cos(At)+1)}{\cos(At)-1}\]

Answer 5

Oh, small mistake. Bleh.
Should be:
\[3\frac{1-\cos^2(At)}{\cos(At)-1}\]
for the middle part, and:
\[3\frac{(1-\cos(At))(1+\cos(At))}{(\cos(At) - 1)} = -3\frac{(\cos(At) - 1)(1+\cos(At))}{(\cos(At) - 1)}\]

Answer 6

thanks :)