So I'm taking a Calculus class, and we're reviewing trigonometry. I was never taught trigonometry, so everything is brand new. MY book is no help at all, so here is the problem:
find an angle between 0 and
[FONT=MathJax_Main][FONT=MathJax_Main][FONT=MathJax_Main][FONT=MathJax_Main]2[FONT=MathJax_Math]π[/FONT][/FONT][/FONT][/FONT][/FONT]
that's equal to
[FONT=MathJax_Main][FONT=MathJax_Main][FONT=MathJax_Main][FONT=MathJax_Main]13[FONT=MathJax_Math]π[/FONT][FONT=MathJax_Main][FONT=MathJax_Main][FONT=MathJax_Main]/[/FONT][/FONT][/FONT][FONT=MathJax_Main]4[/FONT][/FONT][/FONT][/FONT][/FONT]
If you could please respond with lots of detail on how to find this answer. That way I have some idea of what to do. Thank you!
It is important to understand the relationship between radians, degrees, and arc length.
You are most likely familiar with degrees. \(\displaystyle 1\text{°} = 1/360\) of a complete circular rotation (i.e., \(\displaystyle 360\text{°}\)).
Radians give an angle in terms of arc length. \(\displaystyle 1\text{ radian}\) is the angle which segments one radius of arc length. Since the circumference of a circle is equal to \(\displaystyle 2\pi r\), the angle corresponding to a full rotation of a circle is \(\displaystyle 2\pi\text{ radians}\).
Again, \(\displaystyle 2\pi\text{ radians}\) is like saying the angle which would segment \(\displaystyle 2\pi\text{ radii}\) of arc length. Since we know \(\displaystyle 2\pi\text{ radii}\) fit into the complete circle, and \(\displaystyle 360\text{°}\) represents the whole circle, \(\displaystyle 2\pi\text{ radians} = 360\text{°}\).
Just like \(\displaystyle k360\text{°}:k\in\mathbb{Z}\) (That is k*360° where k is an integer) is essentially the same (expect with more rotations) as \(\displaystyle 360\text{°}\),
\(\displaystyle 2k\pi\text{ rad} = 2\pi\text{ rad}:k\in\mathbb{Z}\). This is to say, \(\displaystyle 2\pi\text{ rad}\) is one rotation, \(\displaystyle 4\pi\text{ rad}\) is two rotations, \(\displaystyle 6\pi\text{ rad}\) is three rotations, etc.
So, asking you to find an angle within the interval \(\displaystyle [0,\2pi)\) is like asking you to find an angle within that first rotation that reaches the same point in the circle as the angle that has rotated more than once. For example, if you were asked to find an angle within the interval \(\displaystyle [0,360\text{°})\) that is equal to \(\displaystyle 800\text{°}\), you would subtract some multiple of \(\displaystyle 360\text{°}\) from that angle to make that angle fall within a single rotation. In this case, \(\displaystyle 800\text{°}-2*360\text{°} = 80\text{°}\).
So, in your case, you need to find an angle within the interval \(\displaystyle [0,\2pi)\) that equals \(\displaystyle \frac{13}{4}\pi\) in this case, \(\displaystyle \frac{13}{4}\pi - 2\pi\) will do the trick.
\(\displaystyle \frac{13}{4}\pi - 2\pi = \frac{13}{4}\pi - \frac{8}{4}\pi = \frac{5}{4}\pi\)