Reciprocal, inverse and compound angle trigonometry (HL)

Q: What are the definitions of the three reciprocal trigonometric ratios?

\(\csc \theta = \frac{1}{\sin \theta}\), \(\sec \theta = \frac{1}{\cos \theta}\), and \(\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}\).

Q: What are the two Higher Level Pythagorean identities involving reciprocal ratios?

\(1 + \tan^2 \theta = \sec^2 \theta\) and \(1 + \cot^2 \theta = \csc^2 \theta\).

Q: What is the difference between an inverse trigonometric function and a reciprocal ratio?

An inverse (e.g., \(\arcsin x\)) finds the angle for a given ratio, while a reciprocal (e.g., \(\csc x\)) is simply \(1\) divided by the function's value.

Q: What are the domain and range restrictions for \(y = \arcsin x\)?

Domain: \(-1 \le x \le 1\); Range: \(-\frac{\pi}{2} \le y \le \frac{\pi}{2}\).

Q: What are the domain and range restrictions for \(y = \arccos x\)?

Domain: \(-1 \le x \le 1\); Range: \(0 \le y \le \pi\).

Q: What are the domain and range restrictions for \(y = \arctan x\)?

Domain: \(x \in \mathbb{R}\); Range: \(-\frac{\pi}{2} < y < \frac{\pi}{2}\).

Q: What are the compound angle formulae for sine and cosine?

\(\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B\) and \(\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B\).

Q: What is the double angle identity for tangent?

\(\tan(2A) = \frac{2\tan A}{1 - \tan^2 A}\).

Q: How do the sine, cosine, and tangent functions behave with negative angles (even/odd properties)?

\(\cos(-\theta) = \cos \theta\) (even), while \(\sin(-\theta) = -\sin \theta\) and \(\tan(-\theta) = -\tan \theta\) (odd).

Q: What are the co-function relationships between sine and cosine?

\(\sin(\frac{\pi}{2} - \theta) = \cos \theta\) and \(\cos(\frac{\pi}{2} - \theta) = \sin \theta\).

Mathematics - Analysis and Approaches · IB Diploma Programme (DP) - SL & HL · 4 min read · Updated: Sep 7, 2026

Introduction to Advanced Trigonometry (HL)

Welcome to the Higher Level (HL) extension of trigonometry! In your previous studies, you’ve mastered the basics of sine, cosine, and tangent. Now, we are going to expand your toolkit. We will explore "reciprocal" ratios, look at "inverse" functions that help us find angles, and learn how to break down complex "compound" angles. These tools are essential for the more advanced calculus and vector problems you’ll face later in the course.

Don't worry if this seems like a lot of new symbols at first—they are all just different ways of looking at the same relationships in a triangle!

1. Reciprocal Trigonometric Ratios

In addition to \(\sin \theta\), \(\cos \theta\), and \(\tan \theta\), we define three "reciprocal" ratios. A reciprocal is just \(1\) divided by the original value.

The Big Three Reciprocals

1. Cosecant (\(\csc \theta\) or \(\text{cosec } \theta\)): \(\csc \theta = \frac{1}{\sin \theta}\)
2. Secant (\(\sec \theta\)): \(\sec \theta = \frac{1}{\cos \theta}\)
3. Cotangent (\(\cot \theta\)): \(\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}\)

Memory Trick: Students often mix these up. Notice that the "s" goes with "c" and vice versa: Secant goes with Cosine, and Cosecant goes with Sine.

New Pythagorean Identities

In SL, you learned that \(\sin^2 \theta + \cos^2 \theta = 1\). By dividing this entire equation by either \(\cos^2 \theta\) or \(\sin^2 \theta\), we get two new HL identities:

\(1 + \tan^2 \theta = \sec^2 \theta\)
\(1 + \cot^2 \theta = \csc^2 \theta\)

Quick Tip: These are incredibly useful for simplifying complex expressions or solving equations where you have a mix of \(\tan\) and \(\sec\).

2. Inverse Trigonometric Functions

An inverse function does the opposite of the original function. If \(\sin(30^\circ) = 0.5\), then the inverse function tells us that \(0.5\) came from \(30^\circ\).

Notation and Restrictions

We use the notation \(\arcsin x\), \(\arccos x\), and \(\arctan x\). Because trigonometric functions are periodic (they repeat forever), we have to restrict their range so that we only get one "principal" answer. Without these restrictions, they wouldn't be true functions!

1. \(y = \arcsin x\)
Domain: \(-1 \le x \le 1\)
Range: \(-\frac{\pi}{2} \le y \le \frac{\pi}{2}\)

2. \(y = \arccos x\)
Domain: \(-1 \le x \le 1\)
Range: \(0 \le y \le \pi\)

3. \(y = \arctan x\)
Domain: \(x \in \mathbb{R}\) (all real numbers)
Range: \(-\frac{\pi}{2} < y < \frac{\pi}{2}\)

Common Mistake: Be careful! \(\sin^{-1} x\) (another way to write \(\arcsin\)) is not the same as \(\frac{1}{\sin x}\). The \(-1\) exponent here means "inverse," not "reciprocal." To avoid confusion, IB questions often use the "arc" notation.

3. Compound Angle Identities

Sometimes we need to find the sine or cosine of two angles added together, like \(\sin(A + B)\). You might be tempted to think it's just \(\sin A + \sin B\), but it's not that simple!

The Addition and Subtraction Formulae

\(\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B\)
\(\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B\)
\(\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}\)

Did you know? These formulas allow us to find exact values for angles like \(75^\circ\) by breaking them into \(45^\circ + 30^\circ\). Since we know the exact ratios for \(45\) and \(30\) (see "The unit circle" chapter), we can solve the problem without a calculator!

The Double Angle Identity for Tangent

If we let \(A = B\) in the \(\tan(A + B)\) formula, we get the Double Angle Identity for tangent:
\(\tan(2A) = \frac{2\tan A}{1 - \tan^2 A}\)

Note: Double angle identities for Sine and Cosine are covered in the SL portion of Topic 3.

4. Symmetry and Relationships

Trigonometric graphs have beautiful symmetries. Understanding these helps you solve equations more quickly.

Key Symmetry Properties

Odd and Even Functions:
\(\cos(-\theta) = \cos \theta\) (Cosine is even - it is symmetrical across the y-axis).
\(\sin(-\theta) = -\sin \theta\) (Sine is odd).
\(\tan(-\theta) = -\tan \theta\) (Tangent is odd).

Co-function Relationships

The "co" in cosine stands for "complementary." This is because sine and cosine are related by a \(90^\circ\) (or \(\frac{\pi}{2}\)) shift:
\(\sin(\frac{\pi}{2} - \theta) = \cos \theta\)
\(\cos(\frac{\pi}{2} - \theta) = \sin \theta\)

Summary Checklist

Before moving on to solving complex equations, make sure you can:
1. Recall the three reciprocal ratios (\(\sec, \csc, \cot\)) without mixing them up.
2. Sketch the graphs of \(\arcsin, \arccos,\) and \(\arctan\) and state their domains and ranges.
3. Select the correct compound angle formula from your formula booklet to expand an expression.
4. Use the Pythagorean identities \(1 + \tan^2 \theta = \sec^2 \theta\) to simplify expressions.

Key Takeaway: Trigonometry at HL is all about identities. If an equation looks impossible, there is usually a formula that can "transform" it into a simpler quadratic or a single trigonometric ratio.

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What are the definitions of the three reciprocal trigonometric ratios?

\(\csc \theta = \frac{1}{\sin \theta}\), \(\sec \theta = \frac{1}{\cos \theta}\), and \(\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}\).

What are the two Higher Level Pythagorean identities involving reciprocal ratios?

\(1 + \tan^2 \theta = \sec^2 \theta\) and \(1 + \cot^2 \theta = \csc^2 \theta\).

What is the difference between an inverse trigonometric function and a reciprocal ratio?

An inverse (e.g., \(\arcsin x\)) finds the angle for a given ratio, while a reciprocal (e.g., \(\csc x\)) is simply \(1\) divided by the function's value.

What are the domain and range restrictions for \(y = \arcsin x\)?

Domain: \(-1 \le x \le 1\); Range: \(-\frac{\pi}{2} \le y \le \frac{\pi}{2}\).

What are the domain and range restrictions for \(y = \arccos x\)?

Domain: \(-1 \le x \le 1\); Range: \(0 \le y \le \pi\).

What are the domain and range restrictions for \(y = \arctan x\)?

Domain: \(x \in \mathbb{R}\); Range: \(-\frac{\pi}{2} < y < \frac{\pi}{2}\).

What are the compound angle formulae for sine and cosine?

\(\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B\) and \(\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B\).

What is the double angle identity for tangent?

\(\tan(2A) = \frac{2\tan A}{1 - \tan^2 A}\).

How do the sine, cosine, and tangent functions behave with negative angles (even/odd properties)?

\(\cos(-\theta) = \cos \theta\) (even), while \(\sin(-\theta) = -\sin \theta\) and \(\tan(-\theta) = -\tan \theta\) (odd).

What are the co-function relationships between sine and cosine?

\(\sin(\frac{\pi}{2} - \theta) = \cos \theta\) and \(\cos(\frac{\pi}{2} - \theta) = \sin \theta\).

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