Inverse Trigonometric Functions

Inverse Trigonometric Functions

Inverse Sine Function

The graph of y = sin x does not pass the vertical line test due to the different values of x that have the same y value.  However, if you restrict the doman to the interval -π/2 ≤ x ≤ π/2, it becomes a one-to-one function. 

Now the inverse is possible to graph.  

The domain of y = sin x (-π/2, π/2) becomes the inverse range and the range of y = sin x (-1, 1) becomes the inverse domain.  This is called y = arcsin x.

Inverse Cosine Function

The graph of y = cos x does not pass the vertical line test due to the different values of x that have the same y value.  However, if you restrict the doman to the interval 0 ≤ x ≤ π, it becomes a one-to-one function. 

Now the inverse is possible to graph.

The domain of y = cos x (0, π) becomes the inverse range and the range of y = cos x, (-1, 1), becomes the inverse domain.  This is called y = arccos x.

Inverse Tangent Function

The tangent graph is repetitive and therefore, the domain can be restricted to the interval (-π/2, π/2).  

The domain of y = tan x (-π/2, π/2) becomes the inverse range and the range of y = tan x (-∞, ∞) becomes the inverse domain.

Notice that the vertical asymptotes of the y = tan x graph become the horizontal asymptotes of the y = arctan x graph.