We have discovered that not all functions have an inverse. Only one-to-one functions have inverses. We have also seen that we can salvage pieces and parts of functions by restricting their domain and range. This is necessary for trigonometric functions.

Sine

The Sine function is defined as the vertical coordinate of a point on the unit circle at a given angle. The domain represents angles measured counterclockwise from the positive horizontal axis. The value of \(sin(\theta )\) is the vertical coordinate of the point on the unit circle at the angle \(\theta \). Therefore, the range of sine is \([-1, 1]\).

Graph of \(y = sin(\theta )\).

The sine function is not one-to-one. If we want an inverse, then we’ll have to restrict the domain. The most common restriction is \(\left [ \frac {\pi }{2}, \frac {\pi }{2} \right ]\).

With this restriction, the inverse of sine is called arcsine, abbreviated arcsin.

Note: Of course you can also use the general inverse notation: \(sin^{-1}(x)\).

The range of arcsine now represents angles, just as the domain of sine does.

Because of this restriction, we must keep track of domains and ranges when sine and arcsine interact.

\(\blacktriangleright \) The domain of \(\sin (\theta )\) is \((-\infty , \infty )\), and the range is \([-1,1]\).

\(\blacktriangleright \) The domain of \(arcsin(y)\) is \([-1,1]\), and the range is \(\left [ -\frac {\pi }{2}, \frac {\pi }{2} \right ]\).

Cosine

The Cosine function is defined as the horizontal coordinate of a point on the unit circle at a given angle. The domain represents angles measured counterclockwise from the positive horizontal axis. The value of \(cos(\theta )\) is the horizontal coordinate of the point on the unit circle at the angle \(\theta \). Therefore, the range of cosine is \([-1, 1]\).

Graph of \(y = cos(\theta )\).

The cosine function is not one-to-one. If we want an inverse, then we’ll have to restrict the domain. The most common restriction is \([0, \pi ]\).

With this restriction, the inverse of cosine is called arccosine, abbreviated arccos.

Note: Of course you can also use the general inverse notation: \(cos^{-1}(x)\).

The range of arccos now represents angles, just as the domain of cosine does.

Because of this restriction, we must keep track of domains and ranges when cosine and arccosine interact.

\(\blacktriangleright \) The domain of \(\cos (\theta )\) is \((-\infty , \infty )\) and the range is \([-1,1]\).

\(\blacktriangleright \) The domain of \(arccos(y)\) is \([-1,1]\), but the range is \([ 0, \pi ]\).

Tangent

Arctangent uses the middle piece on \(\left ( -\frac {\pi }{2}, \frac {\pi }{2} \right )\).

With this restriction, the inverse of tangent is called arctangent, abbreviated arctan.

Note: Of course you can also use the general inverse notation: \(tan^{-1}(x)\).

The range of arctan now represents angles, just as the domain of tangent does.

All aspects of a function and its inverse are reversed.

The graph of tangent has vertical asymptotes at \(-\frac {\pi }{2}\) and \(\frac {\pi }{2}\), therefore, The graph of arctangent has horizontal asymptotes at \(-\frac {\pi }{2}\) and \(\frac {\pi }{2}\).