See inverse function notes for a review of inverse functions. In the graph at the left, notice that the sine function, pink and dashed, is not 1-to-1 because it is periodic and repeats every 2. The arcsine of zero is zero, sin -1 0 is 0.
The arccosine of a negative number is a second quadrant angle, cos -1 - is in quadrant II. The arctangent of zero is zero, tan -1 0 is 0. Just as the Sine gives a value for a given angle, the angle for a given value can also be calculated. Arcsin or Inverse Sin is that process. Sin can be defined basically in the context of a right angled triangle.
In a much broader sense, the sin can be defined as a function of an angle, where the magnitude of the angle is given in radians.
It is the length of the vertical orthogonal projection of the radius of a unit circle. In modern mathematics, it is also defined using Taylor series, or as solutions to certain differential equations. These inverse functions have the same name but with 'arc' in front. On some calculators the arcsin button may be labelled asin, or sometimes sin So the inverse of sin is arcsin etc. Use arcsin when you know the sine of an angle and want to know the actual angle.
See also Inverse functions - trigonometry. In the above figure, click on 'reset'.
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