Inverse trig relations

Just like with algebraic graphs, you can find the inverse of trig functions by swapping the x and y values:

For sin(𝜭) = x the inverse is sin(x) = 𝜭

y is the angle whose sine is x

These inverses are called inverse trig relations because they do not have the properties of functions

When solving an inverse trig relation it's best to be clear that all coterminal angles to the solution are also solutions. So, because sin(pi/2) = 1, the inverse would actually be sin(pi/2 + 2npi) = 1

Inverse trig relations are expressed like sin^-1(x) or as arcsin.

The first notation is not actually a negative exponent, so sin^-1(x) != 1/sin(x)

Inverse trig functions

If you limit an inverse trig relation to one period then you have an inverse trig function.

This is notated with a capital letter - Arcsinx

Arccos(1/2) = y

cos is 1/2 when 𝜭 = pi/3 and 𝜭 = 5pi/3

Arccos has the range of [0,pi]

y = pi/3

Here are the domain and ranges of inverse trig functions:

Name:    Domain:                Range:

Arcsin   [-1,1]                 [-pi/2,pi/2]

Arccos   [-1,1]                 [0,pi]

Arctan   [-inf, inf]            (-pi/2, pi/2)

Arccsc   (-inf, -1] U [1, inf)  [-pi/2, 0) U (0, pi/2]

Arcsec   (-inf, -1] U [1, inf)  [0,pi/2) U (pi/2, pi]

Arccot   (-inf, inf)            (0,pi)

Trig functions of inverse trig functions

Because sin and Arcsin are inverses, they have the same relationship as something like sqt(x^2) = (sqt(x))^2 = x or x^lny = ln(x^y) = x

So sin^-1(sinx) = x for x in [0,pi] and sec(Arcsec(x)) = x for x in (-inf, -1] U [1, inf)

You can also build equations by mixing any of 12 trig functions

sec(cot^-1(x)) = sqt(1/x^2 + 1)

I need to practice this…