An expression like
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Created by StemPy LabAn expression with a root or a fraction inside an inverse function often hides a simple angle. Put for , and or for . The expression becomes a trigonometric function of , or . Take the inverse, and then check that the angle is in the branch. If it is not, move it into the branch with the rules of the first course.
To solve an equation with inverse functions, first find the domain. Then use identities or take the sine, cosine or tangent of both sides to remove the inverse functions. Solve the algebra that is left. Taking the tangent or squaring can add roots that do not satisfy the original equation, so always test every root in the original equation. An equation with a single inverse function may have no solution when the other side is outside its range.
Every inverse function brings a condition on its input, and all of them must hold together. The domain of an expression is where all the conditions are true, and it can be a short interval or even one point. The range comes from the branch and from whether the parts rise or fall. A graph of the two sides of an equation, or a scan on the computer, shows how many roots there are and whether an algebra root was false.
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