Inverse Trig Functions

Inverse Trig Functions

In order for a line to have an inverse, it must pass the Horizontal Line Test

y = sinx does not past the horizontal line test on the interval of (-∞, ∞)

y = sinx can be restricted in order for it to pass the horizontal line test and have an inverse

On the interval of [-π, π], the function has an inverse

On the interval of [-π, π]:

y = sinx is increasing

y sinx takes on its full range of values

y = sinx passes the Horizontal Line test

This unique inverse is called the inverse sine function, and it is denotes by

y = arcsinx          or          y =sin⁻¹ x

**It is important to note that y = arcsinx if and only if siny = x**

Similar to y = sinx, y = cosx does not pass the Horizontal Line Test

 

On the interval of [0, π] though:

y = cosx is decreasing

y = cosx takes on its full range of values

 y = cosx does pass the Horizontal Line Test

On the interval of [0, π] cosine does have an inverse, and it is denoted by

This is called the inverse cosine function

y = arccosx            or            y = cos⁻¹x

An inverse tangent function can also be defined by restricting the domain

if the domain of y = tanx is restricted to (-π/2, π/2)