Inverse Trigonometric Functions

We know the trigonometric functions are sine, cosine, tangent, secant, cosecant, and cotangent.
But what are inverse trigonometric functions?

Inverse Trigonometric Functions: the inverse functions of the trigonometric functions with restricted domains.

Which trig functions have inverses?
-sine
-cosine
-tangent
Why?
Because these three trig functions' domains can be restricted so it can be 'one-to-one'. For a function to have an inverse, the original function must be one-to-one. This means for every value of y, there is only one x. An easy way to test if a function's one-to-one, is seeing if the function's graph passes the horizontal line test.

For Example-
We know that sine's graph looks like this:

If you were to draw a horizontal line through the graph, it intersects at more than one point. This means the function is not one-to-one. But it can be 'restricted' so that it's one-to-one.

Sine's restrictions are as follows:
Domain:
Range:

This makes it so that the graph passes the horizontal line test, so it's one-to-one with these restrictions.

Cosine's restrictions:
Domain:
Range:

Tangent's restrictions:
Domain: All real numbers
Range:

For the inverse tangent graph, the (x,y) coordinates are flipped, so (x,y) becomes (y,x)
Ex: (2, 3) becomes (3, 2)

Key Inverse facts to remember:
- The output of any inverse trig function is always an angle:

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- >These are all equivalent to each other.
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