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Inverse Trigonometric Functions

Sin, cos and tan are many-to-one, so to invert them we restrict their domains. The resulting inverse functions arcsin, arccos and arctan each have a specific domain and range you must know.

25 min Video by Zeeshan Zamurred Trigonometric Functions

Edexcel A level Maths: 6.5 Inverse Trigonometric FunctionsWatch the full walkthrough before the notes below.

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What you'll be able to do

Understand why the domains are restricted
State the domain and range of each inverse
Sketch arcsin, arccos, arctan
Evaluate inverse trig values

Restricting the domain

A function only has an inverse if it is one-to-one. We restrict $sin$ to $[- \frac{π}{2}, \frac{π}{2}]$ , $cos$ to $[0, π]$ , and $tan$ to $(- \frac{π}{2}, \frac{π}{2})$ . The inverse swaps domain and range.

arcsin : [- 1, 1] \to [- \frac{π}{2}, \frac{π}{2}]

Domain and range of arcsin.

Domains and ranges

$arcsin x$ : domain $[- 1, 1]$ , range $[- \frac{π}{2}, \frac{π}{2}]$ . $arccos x$ : domain $[- 1, 1]$ , range $[0, π]$ . $arctan x$ : domain all reals, range $(- \frac{π}{2}, \frac{π}{2})$ .

1We need the angle in

[- \frac{π}{2}, \frac{π}{2}]

with sine

\frac{1}{2}

2That is

\frac{π}{6}

Answer

\frac{π}{6}

Tip — arccos has range [0, π] — its answers are never negative.

Formula recap

arcsin x : [- 1, 1] \to [- \frac{π}{2}, \frac{π}{2}]

Inverse sine.

arccos x : [- 1, 1] \to [0, π]

Inverse cosine.

arctan x : R \to (- \frac{π}{2}, \frac{π}{2})

Inverse tangent.

Common mistakes to avoid

Giving arccos a negative answer.

arccos has range [0, π]; answers are in that interval.

Using domain [−1,1] for arctan.

arctan accepts all real numbers.

Key takeaways

Restrict domains to make sin, cos, tan one-to-one.
arcsin/arccos domain [−1,1]; arctan domain all reals.
Ranges: arcsin [−π/2, π/2], arccos [0, π], arctan (−π/2, π/2).

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