• Mar 24th 2009, 01:19 PM
bigfoot
Please can you tell me how to solve the following equations for values of x between (0.2 π) {π is pie}

a) tan (4x) = 1

b) sin(x-2π) + sin (x - 2π) = ½

c) 2 cos (x + ½) = -1

d) sin x + cos x = 3/2
• Mar 24th 2009, 01:41 PM
gosualite
Well when you want to isolate a variable that's inside a trig function, you just take the inverse trig function.

For instance, with the first problem:

$\displaystyle \tan(4x)=1$.

We take $\displaystyle \tan^{-1}$ of both sides (sometimes, $\displaystyle \tan^{-1}$ is written and pronounced $\displaystyle \arctan$). Get:

$\displaystyle \tan^{-1}(\tan(4x)) = \tan^{-1}(1)$, so resolving the left side gives:

$\displaystyle 4x = \tan^{-1}(1)$.

We now have to figure out what $\displaystyle \tan^{-1}(1)$ is. Well it's like asking "what angle in a right triangle gives a ratio of opposite side over adjacent side equal to 1?" It should be obvious that a 45-45-90 triangle passes this test, so $\displaystyle \tan^{-1}(1)$ resolves to $\displaystyle 45^o$, or $\displaystyle \frac{\pi}{4}$. However, there may be more answers. Notice that $\displaystyle \frac{5\pi}{4}$ works as well. In fact, for any integer n, $\displaystyle \tan\left(n\pi + \frac{\pi}{4}\right) = 1$, so $\displaystyle \tan^{-1}(1)$ could be any of these.

We take this info and go back to our equation:

$\displaystyle 4x = \tan^{-1}(1)$. So:

$\displaystyle 4x = n\pi + \frac{\pi}{4}$. Then:

$\displaystyle x = \frac{1}{4}\left(n\pi + \frac{\pi}{4}\right)$.

All right, but you were given a restriction of possible values, $\displaystyle 0 < x < 2\pi$.

So we want to figure out the only values of n that make our solution valid. $\displaystyle 0 < \frac{1}{4}\left(n\pi + \frac{\pi}{4}\right) < 2\pi$,
$\displaystyle 0 < n\pi + \frac{\pi}{4} < 8\pi$,

$\displaystyle -\frac{\pi}{4} < n\pi < 8\pi - \frac{\pi}{4}$,

$\displaystyle -\frac{1}{4} < n < 8 - \frac{1}{4}$.

So n could only be 0,1,2,3,4,5,6, or 7.
• Mar 24th 2009, 03:44 PM
bigfoot
Thanks for your help. I will try to work out the others. :)
• Mar 25th 2009, 02:41 AM
bigfoot
How do we known that 45 degrees is pi over 4?
• Mar 25th 2009, 05:10 AM
stapel
Quote:

Originally Posted by bigfoot
How do we known that 45 degrees is pi over 4?

How does $\displaystyle 2\pi"$ relate to $\displaystyle 180^{\circ}"$?

Dividing by 8, how then does $\displaystyle \frac{\pi}{4}"$ relate to $\displaystyle 45^{\circ}"$?

(Wink)