Confusion with respect to a general angle

Hello,

I shall post the problem with the answer (s) as a given, but it's more of an explanation on what's just happened from which I'm after.

Anyway:

$\displaystyle -540 < 3x < 540 $

$\displaystyle tan\ 3x = -1 \Rightarrow 3x = -405, -225, -45, 135, 315, 495 $

$\displaystyle x = -135, -75 , -15, 45, 105, 165 $

I'm confused how we compute these angles in a problem like this. It's when the coefficient of x is >1 and the range is bigger than $\displaystyle 360^{\circ} $

If I was given a problem like this:

$\displaystyle 0 < x < 360^{\circ} $

$\displaystyle tan\ x = (1) $

For all positive angles - I would have no problem.

I would be grateful for an explanation on what's going on when it's like the preceding case at the beginning of the thread.

Thank you for your attention.

Re: Confusion with respect to a general angle

Quote:

Originally Posted by

**astartleddeer** I'm confused how we compute these angles in a problem like this. It's when the coefficient of x is >1 and the range is bigger than $\displaystyle 360^{\circ} $

Substitute $\displaystyle y = nx, n\in \mathbb{N}$. We will come to deal with this later. So the question is what does angle $\displaystyle |y| < \alpha$ mean when $\displaystyle \alpha > 2 \pi$. Well geometrically when we define, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle, any angle exceeding $\displaystyle 2 \pi$ happens to wrap around to the first quadrant in an 2D plane. Now we start from this and using Pythagoras Theorem define $\displaystyle \sin, \cos, \tan$ and other trigonometric ratios. Now once we start dealing with these as general functions defined on $\displaystyle \mathbb{R}$ we have to deal with any real number which can be $\displaystyle > 2 \pi$. Here we are very specific as to what domain and range we are restricting our function (to be one-one, onto, or both) to make our results or computations meaningful according to the constraints of the problem.

One such example would be digital filter design where we do an operation called phase unwrapping in order to make the phase response a continuous function by suitably accounting for the angular wrapping about $\displaystyle 2\pi$. However all this is from engineering application point of view and we have to have a concrete mathematical definition which holds good in general, hence we extend saying an angle can be any real number and $\displaystyle \sin$, $\displaystyle \cos$, $\displaystyle \tan$...etc are periodic functions (this can be as well developed from algebraic point of view but that would not answer your question).

Once you have extended to this definition of trigonometric function $\displaystyle |y| < \alpha, \alpha > 2\pi$ makes perfect sence and we can say we need $\displaystyle -\frac{\alpha}{n} < x < \frac{\alpha}{n}$ as the required angle $\displaystyle \in \mathbb{R}$.

Re: Confusion with respect to a general angle

You may use:

$\displaystyle \tan\left(135^{\circ} \right)=-1$

$\displaystyle \tan\left(x+k\cdot180^{\circ} \right)=\tan(x)$ where $\displaystyle k\in\mathbb{Z}$

to find all of the solutions within the given range.