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Basic Trigonometric Ratios

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:
    \cos^{-1}x&=&\arccos x \\
    \tan^{-1}x&=&\arctan x
  • 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: