Topic 3.11 · HL only

Reciprocal and Inverse trig functions: notes and practice questions

Summary

Reciprocal trig functions (sec, csc, cot) are defined as reciprocals of cos, sin, and tan, respectively. They have related Pythagorean identities: $1 + \tan^2 x = \sec^2 x$ and $1 + \cot^2 x = \csc^2 x$ . Solving equations involves converting to primary ratios.
Inverse trig functions (arcsin, arccos, arctan) require restricting domains to ensure one-to-one correspondence. Each has a specific domain and principal range.
Derivatives of reciprocal functions include $\frac{d}{dx}(\sec x) = \sec x \tan x$ and $\frac{d}{dx}(\csc x) = -\csc x \cot x$ , etc.
Derivatives of inverse functions are $\frac{d}{dx}(\arcsin x) = \frac{1}{\sqrt{1-x^2}}$ , $\frac{d}{dx}(\arccos x) = -\frac{1}{\sqrt{1-x^2}}$ , and $\frac{d}{dx}(\arctan x) = \frac{1}{1+x^2}$ .
Standard integrals for inverse functions involve forms like $\int \frac{1}{\sqrt{a^2-x^2}} dx = \arcsin(\frac{x}{a}) + C$ and $\int \frac{1}{a^2+x^2} dx = \frac{1}{a} \arctan(\frac{x}{a}) + C$ .

How it is examined

Domain and range of the inverse functions is the assessable fact that is not in the booklet: $\arcsin$ and $\arctan$ return values in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ (open for $\arctan$ ), $\arccos$ in $[0, \pi]$ . Identity manipulation into $\sec$ and $\cot$ is Paper 1. 3 to 6 marks.

Given in the booklet

The two Pythagorean identities are given. The domains and ranges of the inverse functions are not.

Key ideas

Definition of the reciprocal trigonometric ratios $\sec\theta$ , $\csc\theta$ and $\cot\theta$ .
Pythagorean identities: $1 + \tan^2\theta = \sec^2\theta$ ; $1 + \cot^2\theta = \csc^2\theta$ .
The inverse functions $f(x) = \arcsin x$ , $f(x) = \arccos x$ , $f(x) = \arctan x$ ; their domains and ranges; their graphs.

Linking questions

Feeds AHL 5.15 (derivatives and integrals of these functions).
The IB writes cosecant as $\operatorname{cosec}$ , not $\csc$ ; both should be accepted from a student.

Practice questions

5 questions · 3 medium · 2 hard

Showing 5 of 5

Question 1

MediumPaper 1 · no calculator5 marks

Solve the equation $\csc^2 \theta + \cot \theta = 1$ for $-\pi \le \theta \le \pi$ .

Question 2

HardPaper 1 · no calculator5 marks

(a)

Show that $arctan(3) - arctan\left(\frac{1}{2}\right) = \frac{\pi}{4}$ .

[2]

(b)

Hence, or otherwise, find the exact value of $arctan(3) + arctan(2)$ .

[3]

Question 3

MediumPaper 2 · calculator5 marks

Solve the equation $3\tan^2{x} + 4\sec{x} + 1 = 0$ , for $-\pi \le x \le \pi$ .

Question 4

HardPaper 1 · no calculator7 marks

Solve the equation $\arcsin(x) + \arcsin(\sqrt{3}x) = \frac{\pi}{2}$ for $x$ .

Question 5

MediumPaper 2 · calculator7 marks

By using the substitution $x^2 = 3 \sin \theta$ , show that $∫ \frac{x}{\sqrt{9-x^4}} \text{d}x = \frac{1}{2} \arcsin \left( \frac{x^2}{3} \right) + c$ .

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What does Reciprocal and Inverse trig functions cover in IB Maths AA?

Reciprocal trig functions (sec, csc, cot) are defined as reciprocals of cos, sin, and tan, respectively. They have related Pythagorean identities: 1 + tan^2 x = sec^2 x and 1 + cot^2 x = csc^2 x. Solving equations involves converting to primary ratios. Inverse trig functions (arcsin, arccos, arctan) require restricting domains to ensure one-to-one correspondence. Each has a specific domain and principal range. Derivatives of reciprocal functions include (d)/(dx)(sec x) = sec x tan x and (d)/(dx)(csc x) = -csc x cot x, etc.

Is Reciprocal and Inverse trig functions SL or HL?

Reciprocal and Inverse trig functions is HL only. SL students are not examined on it.

How do I revise Reciprocal and Inverse trig functions for IB Maths AA?

Start from the core idea: reciprocal trig functions (sec, csc, cot) are defined as reciprocals of cos, sin, and tan, respectively. They have related Pythagorean identities: 1 + tan^2 x = sec^2 x and 1 + cot^2 x = csc^2 x. Solving equations involves converting to primary ratios. In the exam: domain and range of the inverse functions is the assessable fact that is not in the booklet: arcsin and arctan return values in [-(π)/(2), (π)/(2)] (open for arctan), arccos in [0, π]. Identity manipulation into sec and cot is Paper 1. Then practise exam-style questions, easiest first, writing out every step of your working before you check it.

How does FourtyFive help me practise Reciprocal and Inverse trig functions?

FourtyFive has 5 Reciprocal and Inverse trig functions questions. Every answer you write is marked mark by mark, IB-style, and you see where each mark was won or lost. Every part has a hint, the AI tutor helps you through the step you are stuck on, and your Study Profile picks what to practise next.

Is FourtyFive free for Reciprocal and Inverse trig functions practice?

Yes. A free account gives you 50 marked answers a month, and you do not need a card to sign up.

Can I handwrite Reciprocal and Inverse trig functions answers on an iPad?

Yes. In the FourtyFive iPad app you write your working by hand with Apple Pencil, the way you would on paper, and it is marked the same way.

Reciprocal and Inverse trig functions: notes and practice questions

How it is examined

Linking questions

Practice questions

Question 1

Question 2

Question 3

Question 4

Question 5

Every Reciprocal and Inverse trig functions question, marked for you

Where marks are lost

Practise this topic as a session

Questions,
answered.

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How it is examined

Linking questions

Practice questions

Question 1

Question 2

Question 3

Question 4

Question 5

Every Reciprocal and Inverse trig functions question, marked for you

Where marks are lost

Questions,answered.

Start with the IB questionbank built for you.

Questions,
answered.

Start with the IB question
bank built for you.