Angles in the different quadrants

• By convention, positive angles are measured in the anti-clockwise direction starting from the positive \(x\) axis.

• \(a\) and \(b\) will range between \(-1\) and \(1\) depending on quadrant.
• To find the angle: if \[ \sin \theta =x\]  then \[ \theta = \sin^{-1}x=\arcsin x \]
• For example:
  • \(\sin 30^{\circ} = 0.5\)  Means: The sine of \(30^\circ\) degrees is \(0.5\).
  • \(\arcsin 0.5 = 30^\circ \) Means: The angle whose sine is \(0.5\) is \(30^\circ\) degrees.
• NOTE: \(\displaystyle \sin^{-1}\theta \ne \frac{1}{\sin \theta}\) 
• We also have:
  \begin{eqnarray*}
  \cos^{-1}x&=&\arccos x \\
  \tan^{-1}x&=&\arctan x
  \end{eqnarray*}
• Also note:
  \[ \sin^{2}x=(\sin x)^{2}{\color{red}\neq \sin \left(x^{2}\right)}\]
• The sign of the ratio depends on the quadrant as seen below: