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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: