Inverse Trigonometric Functions
Lessons in this chapter
About Inverse Trigonometric Functions
Inverse Trigonometric Functions covers reversing the trig functions, and the domain restrictions that reversal requires.
Analyzing Domain, Range, and Graphs covers why each trig function needs a restricted domain before it can have a true inverse function, and what that inverse’s graph looks like.
Evaluating Inverse Trigonometric Expressions closes the chapter with finding exact values of inverse trig expressions, including compositions like sin(cos⁻¹(x)).
Both lessons, Analyzing Domain, Range, and Graphs and Evaluating Inverse Trigonometric Expressions, open as practice worksheets worked in the page and scored as you answer. The restricted ranges are the whole difficulty here, and the problems are built so you have to pick the right angle rather than accept the first one a calculator returns. Free to try, with no login needed for your first 3.
Frequently asked questions
Analyzing Domain, Range, and Graphs covers why sine, cosine, and tangent have to be restricted before they can be inverted at all, and what the resulting graphs look like. Evaluating Inverse Trigonometric Expressions then works out exact values, including compositions such as sin(arccos x), where the answer depends on the restricted range rather than on whichever angle comes to mind first. This sits in Functions (F) in the high school Common Core, and in Unit 3 of the AP Precalculus course framework, Trigonometric and Polar Functions.
Each trig function must be restricted to an interval where it’s one-to-one before it can have a true inverse function: otherwise the reversed pairing isn’t a function.
Inverse sine's range is restricted to [-π/2, π/2], and inverse cosine's range is restricted to [0, π]: different intervals, chosen so each covers exactly one full cycle of output values.
An inverse trig function’s graph is the original (restricted) trig function’s graph reflected across the line y = x.
Draw a right triangle representing the inner inverse trig expression, then use that triangle’s sides to evaluate the outer trig function directly.
Forgetting the restricted range and giving an answer outside it: an inverse trig function can only output values from its own specific restricted range, never the full set of angles that share that trig ratio.