# trigonometric equation solve

• Apr 30th 2008, 01:46 PM
sux@math
trigonometric equation solve
Are solving & proving the same thing? My book has a method for "proving" using either the quotient identity or pythagorean identity - you know that whole, LS=RS stuff.

Earlier, though there is a method for "solving" by "let k=cos x" for example.

Well, for some of my assignment, I used the "solving" method *because the question told me to solve* and when it was returned to me, the teacher's notes were:

your factoring is rushed and solving is based on two identities: the tangent and sin^2x+cos^2x=1. You will be able to solve all by using these two.

I don't know what is going on.
If anyone can point me to a nice basic thread on MHP/website on this stuff I will be ever-so grateful!
• Apr 30th 2008, 01:59 PM
icemanfan
Quote:

Originally Posted by sux@math
Are solving & proving the same thing? My book has a method for "proving" using either the quotient identity or pythagorean identity - you know that whole, LS=RS stuff.

Earlier, though there is a method for "solving" by "let k=cos x" for example.

Well, for some of my assignment, I used the "solving" method *because the question told me to solve* and when it was returned to me, the teacher's notes were:

your factoring is rushed and solving is based on two identities: the tangent and sin^2x+cos^2x=1. You will be able to solve all by using these two.

I don't know what is going on.
If anyone can point me to a nice basic thread on MHP/website on this stuff I will be ever-so grateful!

Suppose I asked you to solve the equation $\displaystyle 2 \sin{^2}{x} + 3 \cos{^2}{x} = 3$.

You would have to use the identity $\displaystyle \sin{^2}{x} + \cos{^2}{x} = 1$.

By example:
$\displaystyle 2 \sin{^2}{x} + 3 \cos{^2}{x} = 3$

$\displaystyle 2 \sin{^2}{x} + 2 \cos{^2}{x} + \cos{^2}{x} = 3$

$\displaystyle (2 \sin{^2}{x} + 2 \cos{^2}{x}) + \cos{^2}{x} = 3$

$\displaystyle 2 + \cos{^2}{x} = 3$

$\displaystyle \cos{^2}{x} = 1$ etc.

To prove the identity we used, we would have to employ the definition of the sine and cosine. But the difference between proving and solving is that proving is illustrating that a certain statement is always true, such as $\displaystyle \sin{^2}{x} + \cos{^2}{x} = 1$, whereas solving involves finding the values for the variables that satisfy a given equation. For the above, we have $\displaystyle \cos{x} = -1$ or $\displaystyle \cos{x} = 1$, giving us the answers $\displaystyle x = -\pi + 2\pi{n}$ and $\displaystyle x = 2\pi{n}$.