Before the midterm

Ignoring the range of an inverse trigonometric function

The habit: arcsin and arctan return angles between −π/2 and π/2 and arccos returns angles between 0 and π, so arcsin(sin θ) equals θ only when θ already lies in that range.

4 practice questions are built to catch it, in 1 topic.

Practise against it: 3 questions

What it looks like

A question from implicit differentiation and inverse trigonometric functions, with the answer this slip gives beside the right one.

Introductory2 marks · 2 min

Find the exact value of $\arctan\!\left(\tan\dfrac{3\pi}{4}\right)$ .

With the slip: $\displaystyle \approx 2.356$
Worked through: $\displaystyle \arctan\!\left(\tan\frac{3\pi}{4}\right) = -\frac{\pi}{4}$

3π/4 lies outside (−π/2, π/2), where arctan takes its values. Tangent repeats every π: shift 3π/4 by π into that interval.

See the worked solution

Where it comes up

Implicit differentiation and inverse trigonometric functions4 questions

Practise against it

Each of these has an answer this slip gives. Work them on paper: if you fall into it, the verdict names it and the question comes back in review.

01
Find the exact value of $\arcsin\!\left(\sin\dfrac{5\pi}{6}\right)$ .
Implicit differentiation and inverse trigonometric functionsIntroductory
02
Find the exact value of the expression below. $\displaystyle \cos\bigl(\arctan(-2)\bigr)$
Implicit differentiation and inverse trigonometric functionsStandard
03
Let $g(x) = \arctan\!\left(\dfrac{1}{x}\right)$ for $x \neq 0$ . Each graph below is drawn with dashed lines at $y = \dfrac{\pi}{2}$ and $y = -\dfrac{\pi}{2}$ . Which one is the graph of $g$ ?
Implicit differentiation and inverse trigonometric functionsChallenging

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