Solving for Theta: Understanding the Last Step

[Checkfate]

I don't really understand a step in my textbook, and I think it is better that I understand why they are doing it this way instead of just brushing it off and it coming back to haunt me later!

The question is : Solve for \Theta, 0\leq\Theta\leq2\pi

2\tan\Theta-2=0

I get the right answer, but their last step confuses me...

This is how they show their work :

2\tan\Theta=2

\tan\Theta=1

\Theta=\frac{\pi}{4} I don't fully grasp this step...

I understand that tan is 1 at \frac{\pi}{4} since sin and cosine are both \frac{\sqrt{3}}{2} at that point. But how can you call tan pi/2 when it also is 1 at \frac{5\pi}{4}... Can someone explain what they are doing? They don't even give \frac{5\pi}{4} as an answer in their equation, they just state it later in a sentence after the question.

Thanks..

 

[Checkfate]

Nevermind I understand now, fairly simple. Just convert the equation to \Theta=tan^{-1}(1)

\Theta=\frac{\pi}{4}

This trigonometry stuff is slowly eating my brain, lol. ahh!

 

[homology]

I guess I don't understand. \frac{5\pi}{4} and \frac{\pi}{4} both work, they are both solutions to the equation 2tan\theta -2 = 0. Reducing the equation to \theta=tan^{-1}(1) isn't the answer since you have to pick which part of tangent you want to invert. Putting tan^{-1}(1) into your calculator will give you \frac{\pi}{4} but this is not the only answer. It would be if your inequality read something like 0\leq \theta \leq \frac{\pi}{2} but it doesn't. You're looking for all solutions between zero and 2\pi. I'm not sure why your book doesn't include both solutions though, that's odd. But you were right to be bothered by this.

 

[semc]

hmm...\frac{\pi}{4} is your alpha.after getting your alpha,you are suppose to use it in the general solution and for tan,it is n{pi}+\frac{\pi}{4} where n is all real number.since the question ask for \Theta, 0\leq\Theta\leq2\pi, the only 2 are \frac{5\pi}{4} and \frac{\pi}{4} coz your n can only take 0 and 1.get it?

 

[Checkfate]

I guess that makes sense I just thought that they should show that in their algebraic answer, but what you are saying makes sense and seems to coincide with their instructions. Thanks