Egwald Mathematics: Trigonometry
Elmer G. Wiens
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Natural trig functions | Inverse trig functions
Inverse Natural Trigonometric Functions
The natural trigonometric functions, like sine and cosine, yield a number from an angle. Thus, for z = sin(alpha), if the angle alpha = pi/6, then .5 = sin(alpha). Suppose, now, we want to find the angle, alpha, such that sin(alpha) = z = 1. Then alpha = pi/2, since sin(pi/2) = 1. Once we know alpha, we can find the values of x and y that produce the angle alpha on the circle on the natural trig page functions.
Since we can do this for any angle, we can define a new function called the inverse sine or "arcsin" function by alpha = arcsin(z). For convenience, let us re-label the variables alpha and z, by y and x. (Don't confuse these new y and x labels with the ones on the circle that locate the point P).
Furthermore, we can define inverse functions for each of the natural trigonometric functions: arcsin, arcos, arctan, arccse, arcsec, and arccot.