Year 12 Maths Extension 1 (2027) Trigonometric functions

Graphs of inverse trig functions

Q: What are the domain and range of the inverse trig graphs?

Arcsin: domain minus one to one, range minus pi over 2 to pi over 2. Arccos: domain minus one to one, range zero to pi. Arctan: domain all reals, range minus pi over 2 to pi over 2.

Q: Which inverse trig functions are odd?

Arcsin and arctan are odd, symmetric about the origin. Arccos is neither odd nor even.

Q: Why does arctan have horizontal asymptotes?

Its range is minus pi over 2 to pi over 2, so as x tends to plus or minus infinity the graph approaches y equals plus or minus pi over 2 without reaching them.

Q: What are the endpoints of arcsin and arccos?

Arcsin: minus one, minus pi over 2 and one, pi over 2. Arccos: minus one, pi and one, zero.

Q: Is arccos odd or even?

Neither. Arccos has point symmetry about zero, pi over 2, which satisfies arccos of minus x equals pi minus arccos x.

20 practice questions 2 video lessons Theory + worked examples

Explore the graphs of inverse trigonometric functions in NSW Year 12 Mathematics Extension 1. Each inverse graph is the reflection of a restricted sine, cosine or tangent curve in the line y = x.

You will learn to sketch the graphs of arcsin, arccos and arctan, identify their domain, range and any asymptotes, and classify them as odd, even or neither — building the visual understanding assessed in Extension 1.

Practice 20 questions

Practice questions

Every question with a fully worked solution.

Start practising

Watch 2 video(s)

Graphing Inverse Trig Functions (Arcsin, Arccos, Arctan Explained!) Watch
【PRE-CALCULUS】Graphing Inverse Trigonometric Functions Watch

Theory

Each inverse trig graph is the reflection of a restricted trig graph in $y = x$ . This NSW Year 12 Mathematics Extension 1 topic (NESA outcome ME1-12-03) covers the shape, domain, range, endpoints, asymptotes and symmetry of $y = \sin^{- 1} x$ , $y = \cos^{- 1} x$ and $y = \tan^{- 1} x$ .

Each inverse trig graph is the reflection of a restricted trig graph in the line $y = x$ . Reflecting swaps the axes, so the domain and range of the restricted function become the range and domain of the inverse.

$y = \sin^{- 1} x$ increases from $(- 1, - \frac{π}{2})$ to $(1, \frac{π}{2})$ through the origin, and is odd. $y = \cos^{- 1} x$ decreases from $(- 1, π)$ to $(1, 0)$ and is symmetric about $(0, \frac{π}{2})$ (neither odd nor even). $y = \tan^{- 1} x$ increases through the origin with horizontal asymptotes $y = \pm \frac{π}{2}$ , and is odd.

NESA link. Outcome ME1-12-03 (with MAO-WM-01). The syllabus asks students to graph each restricted trig function, reflect it in $y = x$ to obtain the inverse, and classify $\sin^{- 1} x$ , $\cos^{- 1} x$ and $\tan^{- 1} x$ as odd, even or neither.

Reflecting restricted

y = \sin x

y = x

gives

y = \sin^{- 1} x

y = \tan^{- 1} x

rises through the origin toward the asymptotes

y = \pm \frac{π}{2}

Reflecting in $y = x$ sends each endpoint $(a, b)$ of the restricted trig graph to the endpoint $(b, a)$ of the inverse.

Domain, range, trend and symmetry

	$\sin^{- 1} x$	$\cos^{- 1} x$	$\tan^{- 1} x$
Domain	$[- 1, 1]$	$[- 1, 1]$	all $x$
Range	$[- \frac{π}{2}, \frac{π}{2}]$	$[0, π]$	$(- \frac{π}{2}, \frac{π}{2})$
Trend	increasing	decreasing	increasing
Symmetry	odd	about $(0, \frac{π}{2})$	odd

arcsin odd, arccos symmetric about (0, pi/2), arctan odd with asymptotes y = plus/minus pi/2

Key features. Only $\tan^{- 1}$ has horizontal asymptotes; $\sin^{- 1}$ and $\cos^{- 1}$ have endpoints instead. $\cos^{- 1}$ is neither odd nor even.

How to graph an inverse trig function by reflection

Draw the restricted trig graph. Sine on $[- \frac{π}{2}, \frac{π}{2}]$ , cosine on $[0, π]$ , or tangent on $(- \frac{π}{2}, \frac{π}{2})$ .
Mark the key points — the endpoints (or asymptotes) and the intercepts.
Reflect in $y = x$ . Swap the coordinates of every marked point: $(a, b) \to (b, a)$ .
Join smoothly, keeping the trend and stating the domain, range, and any asymptotes.

Example 1 — Features of

\sin^{- 1} x

State the domain, range, endpoints and symmetry of

y = \sin^{- 1} x

Solution

Domain $[- 1, 1]$ ; range $[- \frac{π}{2}, \frac{π}{2}]$ .

Endpoints $(- 1, - \frac{π}{2})$ and $(1, \frac{π}{2})$ .

It is odd — symmetric about the origin.

Example 2 — Intercepts of

\cos^{- 1} x

Find the

x

- and

y

-intercepts of

y = \cos^{- 1} x

Solution

$x$ -intercept $(y = 0)$ : $\cos^{- 1} x = 0 \Rightarrow x = 1$ , giving $(1, 0)$ .

$y$ -intercept $(x = 0)$ : $\cos^{- 1} 0 = \frac{π}{2}$ , giving $(0, \frac{π}{2})$ .

Intercepts $(1, 0)$ and $(0, \frac{π}{2})$ .

Example 3 — Asymptotes of

\tan^{- 1} x

State the asymptotes of

y = \tan^{- 1} x

and explain why they occur.

Solution

The asymptotes are $y = \frac{π}{2}$ and $y = - \frac{π}{2}$ .

The range is $(- \frac{π}{2}, \frac{π}{2})$ , so as $x \to \pm \infty$ the graph approaches these values but never reaches them.

Example 4 — A point on the graph

The point

(a, \frac{2 π}{3})

lies on

y = \cos^{- 1} x

. Find the exact value of

a

Solution

$\cos^{- 1} a = \frac{2 π}{3}$ , so $a = \cos \frac{2 π}{3}$ .

a

=

\cos \frac{2 π}{3} = - \frac{1}{2}

a = cos(2pi/3) = -1/2

So $a = - \frac{1}{2}$ .

Common pitfalls

Swapping coordinates. Reflecting in

y = x

swaps

x

and

y

: the endpoints of the restricted trig graph become the endpoints of the inverse.

Asymptotes vs endpoints. Only

\tan^{- 1}

has horizontal asymptotes

(y = \pm \frac{π}{2})

;

\sin^{- 1}

and

\cos^{- 1}

stop at endpoints.

Calling arccos odd.

\cos^{- 1}

is neither odd nor even — it is symmetric about

(0, \frac{π}{2})

Range height.

\sin^{- 1}

and

\tan^{- 1}

span a height of

π

(from

- \frac{π}{2}

\frac{π}{2}

);

\cos^{- 1}

spans

0

π

Frequently asked questions

How do you graph arcsin, arccos and arctan?

Graph the trig function on its restricted domain, then reflect that piece in the line $y = x$ . The reflection is the inverse graph.

What are the domain and range of the inverse trig graphs?

$\sin^{- 1}$ : domain $[- 1, 1]$ , range $[- \frac{π}{2}, \frac{π}{2}]$ ; $\cos^{- 1}$ : domain $[- 1, 1]$ , range $[0, π]$ ; $\tan^{- 1}$ : domain all reals, range $(- \frac{π}{2}, \frac{π}{2})$ .

Which inverse trig functions are odd?

$\sin^{- 1}$ and $\tan^{- 1}$ are odd (symmetric about the origin). $\cos^{- 1}$ is neither odd nor even.

Why does arctan have horizontal asymptotes?

Its range is $(- \frac{π}{2}, \frac{π}{2})$ , so as $x \to \pm \infty$ the graph approaches $y = \pm \frac{π}{2}$ without reaching them.

What are the endpoints of arcsin and arccos?

$\sin^{- 1}$ : $(- 1, - \frac{π}{2})$ and $(1, \frac{π}{2})$ . $\cos^{- 1}$ : $(- 1, π)$ and $(1, 0)$ .

Is arccos odd or even?

Neither. $\cos^{- 1} x$ has point symmetry about $(0, \frac{π}{2})$ , which satisfies $\cos^{- 1} (- x) = π - \cos^{- 1} x$ .

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Definitions of inverse trig functions (domain/range)

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Properties, composite functions & transformations

Graphs of inverse trig functions

📖 Theory

Domain, range, trend and symmetry

How to graph an inverse trig function by reflection

Common pitfalls

Frequently asked questions

Theory