Step By Step Calculus » 5.1 - Angles and The Unit Circle

show_hiddenExpand All [+]
Synopsis
Unit circle and Angles: The point with coordinates \left(x,y\right) \left(x,y\right) is on the unit circle centered at origin if and only if
\displaystyle x^2+y^2=1.
\displaystyle x^2+y^2=1.
The unit circle is of central importance for trigonometry. The number \pi \pi is the length of half its circumference. The displacement, i.e., signed distance, along the circle from the point (1,0) (1,0) to another point on the circle is by definition the angle between the two points in radians. Rotation counterclockwise from the positive x x-axis gives a positive angle, rotation clockwise gives a negative angle.

Coterminal Angles: Two angles are coterminal if they have the same terminal sides when their initial sides are placed on the positive x x-axis. For an angle \theta\theta, the coterminal angles are exactly the members of the set of angles {\cal C}=\{\theta+k2\pi\;|\; k\in\mathbb{Z}, k\ne 0\}{\cal C}=\{\theta+k2\pi\;|\; k\in\mathbb{Z}, k\ne 0\}.
Complementary and Supplementary Angles: Two positive angles are complementary if their combined rotation (when you first rotate by one angle and then continue rotating by the other from there) produces a right angle, that is an angle of radian measure \dfrac{\pi}{2}\dfrac{\pi}{2}. Two positive angles are supplementary if their combined rotation produces a straight line, that is an angle of radian measure \pi\pi.
\displaystyle \theta\displaystyle \theta in radians\displaystyle = \frac{\pi}{180}\theta \displaystyle = \frac{\pi}{180}\theta in degrees\displaystyle . \displaystyle .
Classification of Angles: Angles can be classified and given the following names based on their degree or radian measures.
\newcommand{\T}{\rule{0pt}{3.6ex}} \newcommand{\B}{\rule[-2.5ex]{0pt}{0pt}} \begin{tabular}{r|c|c}\hline Angles & Degree Measure & Radian Measure \\ \hline \T\B$ $Right Angle \par & $ \dfrac{1}{4}\cdot 360^\circ=90^\circ$ \par & $ \dfrac{\pi}{2}$ radians \par \\\hline\T\B $ $Acute Angle \par & $ $Any angle in $ \left(0^\circ,90^\circ\right)$ \par & $ $Any angle in $\left(0,\dfrac{\pi}{2}\right)$ \par \\\hline\T\B $ $Straight Angle \par & $ 180^\circ$ \par & $ \pi$ radians \par \\\hline\T\B $ $Obtuse Angle \par & $ $Any angle in $ \left(90^\circ,180^\circ\right)$ \par & $ $Any angle is $\left(\dfrac{\pi}{2},\pi\right)$ \par \\\hline \end{tabular}
Arc Length Given a circle of radius rr, the length of the arc of this circle subtended by an angle \theta\theta (in radians) is given by
\displaystyle l=r\theta.
\displaystyle l=r\theta.