The graph of y = sin x can be visuallised in the figure below:

Domain: all reals Range: [-1,1] Period: 2π Y-intercept: (0,0)

When you restrict the domain of sin x to the interval –π/2 ≤ x ≤ π/2, the following properties should hold:

**1.***y* = sin *x* is an increasing function.

**2.***y* = sin *x *utilizes full range of values, –1 ≤ sin *x ≤ *1.

**3.***y* = sin *x *is a one-to-one function.

In conclusion, on the restricted domain –π/2 ≤ x ≤ π/2, *y* = sin*x *obtains a unique inverse function called the inverse sine function: *y *= arcs in *x *or* y *= sin^{–1}*x*. The inverse function is equivalent to the analogy of an inverse function notation *f *^{–1}(*x*) which is not the same as (f)^{-1}.

Using the domain: [-π/2 ;π/2] let’s plot the graph *y* = arcs in *x*. Remember that y = arcs in x can be written as sin *y* = *x*

**The graph of y = arcsin x **

Domain: $[$-1, 1$]$ Range: $[-\frac{\pi }{2},\frac{\pi }{2}]$

Similarly, we can use 1-1 cos x values to plot the graph of arccos x. The cosine function is a decreasing function and it forms one-to-one values on the interval 0 ≤ x ≤ π, as shown in the figure below:

On this interval, the cos x has an inverse function denoted by y= arccos*x *or* y *= cos^{–1}*x*. Because *y *= arccos*x *and* x *= cos *y *mean the same for 0 ≤ y ≤ π, their graphs are the same, and can be confirmed by the following table of values.

**The graph of y = arccos x**

Domain: [-1, 1] Range: [0, π] Period: Y-int: [0, 0]

The other trigonometric functions require similar restrictions on therefor generating an inverse function.The domain of the section of the tangent that generates the arctan is $\frac{\u2013\pi}{2},\frac{\pi}{2}$ and the inverse tangent function is denoted by* y *= arctan *x *or *y *= tan ^{–1}*x*. Since *y *= arctan *x *and *x *= tan *y *are the same for –2 <*y*<2 their graphs are the same, and can be confirmed by the following table of values.

Domain: $[-\infty , \infty $$]$ Range: $[$$\frac{-\pi }{2}, \frac{\pi }{2}$$]$