Inverse Trigonometric Functions

If , what is ? It could be , or , or , or infinitely many other angles, because sine repeats. The inverse trigonometric functions answer the question with one agreed angle each time. They’re the tools for finding an angle from a ratio, and in calculus they reappear with surprisingly simple derivatives that make them essential in integration.

Why the Domain Has to Be Restricted

A function only has an inverse if it’s one-to-one: each output comes from a single input. Sine fails that badly.

The fix is to keep just one piece of the curve that takes every value once. For sine, the standard choice is . The inverse is then defined by

Cosine and tangent get their own intervals, chosen so that the inverse is continuous and covers every value.

Why doesn’t cosine use the same interval as sine? On , cosine is never negative and it takes most values twice, once on each side of zero. On it falls steadily from 1 to , hitting every value exactly once, so that’s the interval used for arccos. Tangent uses the open interval , because it’s undefined at the endpoints.

The Three Main Inverse Trig Functions

The domain of arcsin and arccos is because sine and cosine never go outside that range: is undefined, since no angle has a sine of 2. The “arc” names come from the unit circle, where the angle in radians is the length of an arc.

Graphs of the Inverse Trig Functions

Each graph is the reflection of the restricted trig graph across the line , like any inverse. Arcsin rises from to . Arccos falls from to . Arctan is defined for every and levels off toward the horizontal asymptotes , which is why it shows up in limits at infinity.

How to Evaluate Inverse Trig Functions

Ask: which angle in the range has this sine, cosine or tangent? Use the exact values from the trigonometric functions lesson.

Example 1: Exact Values

Both answers lie in , as they must.

Example 2: Negative Inputs

: arccos returns angles in . Cosine is at in that range, so the answer is .

: arctan returns angles in , so the answer is , not .

Negative inputs are where the ranges matter most: arcsin and arctan give negative angles for negative inputs, while arccos gives an angle in the second quadrant.

Compositions

Undoing Each Other, With a Catch

for every in . But the other order only works inside the restricted range:

because arcsin can only return angles between and .

Mixing Functions With a Triangle

To find something like , name the angle and draw it.

Let , so : opposite 3, hypotenuse 5. The Pythagorean theorem gives the adjacent side as 4, so . The same method with a variable gives a formula that’s used constantly in calculus: .

Inverse Trig Functions on a Calculator

The , and keys return exactly the angles in the ranges above, so they never give the “other” solution. In RAD mode, ; in DEG mode the same key gives . Both are the same angle. For exact values like , work from the special-angle table instead: a calculator shows 0.5236, and recognising that as takes an extra step.

The Other Three Inverse Functions

Arcsecant, arccosecant and arccotangent exist too, though you’ll meet them far less often. They’re usually defined through the main three, for example

so . Different textbooks choose slightly different ranges for these three, so check the convention your course uses before relying on one.

Inverse Trig Functions in Calculus

Their derivatives are algebraic, with no trig left in them:

Reading those backwards is what makes them so useful: an integral of has the answer , even though nothing in the question looks like trigonometry. That’s the “integration of inverse trig functions” topic in Calculus 2. For finding angles in right triangles numerically, the inverse tan calculator does the arctan step and shows the working.

Common Mistakes to Avoid

• Reading as . That’s .
• Giving an answer outside the range. is , not .
• Assuming always. It’s only true for in .
• Taking arcsin of a number outside . There’s no answer.
• Using degree mode. In calculus, inverse trig answers are in radians.

Practice Problems

Work each one before opening the answer.

Problem 1. Evaluate .

Problem 2. Evaluate .

Problem 3. Evaluate .

Problem 4. Evaluate .

Problem 5. Evaluate .

Problem 6. Find .

Inverse trig functions give one angle. Solving a trig equation means finding all of them, which is the last lesson of this review chapter.