We often have to find analytic expressions for the trigonometric formulaeThe method is illustrated in the following examples.
Example: Find an expression for
Ifthen
Multiply both sides byto obtain
Now multiply both sides by 2 obtainingand subtract
from both sides to obtain
This is a quadratic expression inso we can solve it using the ordinary quadratic formula.
- remember that our quadratic is in terms of
Now take the natural logarithms of both sides to obtain
Since
Example Find an expression for
Ifthen
so
Multiply both sides byto obtain
Now multiply both sides by 2 obtainingand subtract
from both sides to obtain
This is a quadratic expression inso we can solve it using the ordinary quadratic formula.
- remember that our quadratic is in terms of
Now take the natural logarithms of both sides to obtain
Since
Example Find an expression for
Ifthen
so
Multiply both sides byto obtain
Expand the brackets:and move the
term to the left, and the
term to the right, obtaining
Now factorise with
Divide by
Now take the natural logs of both sides and divide by 2.