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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:
\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:
