How do I know whether to use degrees or radians?

The question will always specify the interval for the solution. If it's given in degrees (e.g., $0^\circ \le x \le 360^\circ$), your answer must be in degrees. If it's given in radians (e.g., $0 \le x \le 2\pi$), your answer must be in radians. Make sure your calculator is in the correct mode!

There are so many identities. How do I know which one to use?

Look for clues. If an equation has different trig functions (like $\sin\theta$ and $\cot\theta$), use an identity to write everything in terms of one function. If it has different angles (like $2\theta$ and $\theta$), use a double angle formula. If you see a sum like $a\sin\theta + b\cos\theta$, think of the harmonic form.

What is the point of the harmonic form?

It simplifies expressions. An expression like $3\sin x + 4\cos x$ is difficult to analyse. By converting it to $5\sin(x+53.1^\circ)$, you can immediately see its amplitude (maximum value) is 5, and you can easily solve equations like $3\sin x + 4\cos x = 2$.

I've found the principal value from my calculator. How do I find all the other solutions?

Use the symmetry of the trigonometric graphs or a CAST diagram. For an angle $\alpha$: - For $\sin x = k$: the other solution is $180^\circ - \alpha$ (or $\pi - \alpha$). - For $\cos x = k$: the other solution is $360^\circ - \alpha$ (or $2\pi - \alpha$, or simply $-\alpha$). - For $\tan x = k$: the other solution is $180^\circ + \alpha$ (or $\pi + \alpha$). Then, add or subtract multiples of $360^\circ$ (or $2\pi$) to find all solutions within the required range.

Where can I learn Trigonometry for Cambridge 9709?

This free lesson on MarkScheme covers syllabus point 2.3 (Trigonometry) for Cambridge A-Level Mathematics. It includes visual revision, key points, and links to real past-paper questions with mark-scheme marking at markscheme.app/courses/9709/2-3-trigonometry.

Is Trigonometry (2.3) on 9709 free to study?

Yes. The Mathematics 9709 course on MarkScheme is 100% free — including premium visual lessons, exam tips, and past-paper marking. No paywall for Trigonometry.

Trigonometry 9709 — Free A-Level Maths

Trigonometry

Welcome to Pure Mathematics 2 Trigonometry. This topic significantly expands on your P1 knowledge, introducing a new set of identities and powerful techniques. We will explore reciprocal functions, double angle formulae, and the harmonic form, all of which are essential tools for solving more advanced trigonometric equations that frequently appear in examinations.

BY THE END, YOU CAN…

1Recall and use trigonometric identities, including reciprocal functions ($sec$, $csc$, $cot$) and the Pythagorean identities.
2Use compound and double angle formulae to manipulate and simplify trigonometric expressions.
3Express expressions of the form $a\sin\theta + b\cos\theta$ in the harmonic form $R\sin(\theta \pm \alpha)$ or $R\cos(\theta \pm \alpha)$ .
4Solve trigonometric equations for a specified interval, finding all possible solutions.

In simple terms

A friendly intro before the formal notes — no formulas yet.

Trigonometry Toolkit

Master trigonometry by learning how fundamental identities combine to solve complex equations. We'll build from basic rules to powerful techniques for simplifying and solving.

THINK OF IT LIKE…

Think of trigonometry as a mechanical toolkit. The basic identities like $\sin^2\theta + \cos^2\theta = 1$ are your standard spanners. The double angle and harmonic form formulae are like specialised power tools that let you transform a complex problem (a stubborn, oddly-shaped bolt) into a simple one you can solve easily.

1
Start with the fundamental identities $\sin^2\theta + \cos^2\theta = 1$ and $\tan\theta = \sin\theta / \cos\theta$ , which arise from the unit circle.
2
Combine two trigonometric terms into one using the harmonic form $a\sin\theta + b\cos\theta = R\sin(\theta + \alpha)$ , where $R = \sqrt{a^2+b^2}$ .
3
Simplify expressions involving different angles using the double angle formulae, like $\sin(2\theta) = 2\sin\theta\cos\theta$ , derived from compound angle rules.
4
To solve an equation, use identities to simplify, find the principal solution, then use the graph's symmetry to find all other solutions in the required interval.

Explore the concept

Use the live diagram and synced steps — play it or tap a step card to walk through.

Key formulas

Tap any symbol to reveal exactly what it means and its units.

$\sec\theta = \frac{1}{\cos\theta} \quad \csc\theta = \frac{1}{\sin\theta} \quad \cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta}$

Tap a symbol — great for exam definitions

$1 + \tan^2\theta \equiv \sec^2\theta \\ 1 + \cot^2\theta \equiv \csc^2\theta$

Tap a symbol — great for exam definitions

$\sin(2A) \equiv 2\sin A \cos A \\ \cos(2A) \equiv \cos^2 A - \sin^2 A \equiv 2\cos^2 A - 1 \equiv 1 - 2\sin^2 A \\ \tan(2A) \equiv \frac{2\tan A}{1 - \tan^2 A}$

Tap a symbol — great for exam definitions

$a\sin\theta + b\cos\theta$

Tap a symbol — great for exam definitions

Full topic notes

Formal explanation with the rigour you need for the exam.

Reciprocal and Pythagorean Identities

Beyond sine, cosine, and tangent, there are three reciprocal trigonometric functions: secant (sec), cosecant (csc), and cotangent (cot). These are simply the reciprocals of the main three and are crucial for simplifying a wider range of expressions.

$\sec\theta = \frac{1}{\cos\theta} \quad \csc\theta = \frac{1}{\sin\theta} \quad \cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta}$

By dividing the fundamental identity $\sin^2\theta + \cos^2\theta \equiv 1$ by $\cos^2\theta$ and $\sin^2\theta$ respectively, we can derive two further essential Pythagorean identities. You must be able to recall and use these fluently.

$1 + \tan^2\theta \equiv \sec^2\theta \\ 1 + \cot^2\theta \equiv \csc^2\theta$

Double Angle Formulae

The double angle formulae are a special case of the compound angle formulae (found in the MF19 formula booklet) where the two angles are equal, i.e., $A=B$ . Their main purpose is to change the argument of a trigonometric function, typically from $2A$ to $A$ . This is fundamental for solving equations that contain a mix of angles, like $\cos(2\theta)$ and $\sin\theta$ .

$\sin(2A) \equiv 2\sin A \cos A \\ \cos(2A) \equiv \cos^2 A - \sin^2 A \equiv 2\cos^2 A - 1 \equiv 1 - 2\sin^2 A \\ \tan(2A) \equiv \frac{2\tan A}{1 - \tan^2 A}$

Use $\cos(2A) = 2\cos^2 A - 1$ when you want to eliminate other trig functions and form a quadratic in $\cos A$ .
Use $\cos(2A) = 1 - 2\sin^2 A$ when you want to form a quadratic in $\sin A$ .
Use $\sin(2A) = 2\sin A \cos A$ to combine or separate terms, often leading to factorisation.

EXAM TIP

A very common exam question involves an equation with both $\cos(2\theta)$ and a term in $\cos\theta$ or $\sin\theta$ . This is a strong signal to immediately substitute the appropriate double angle formula for $\cos(2\theta)$ to create a quadratic equation.

Harmonic Form: $a\sin\theta + b\cos\theta$

Expressions of the form $a\sin\theta + b\cos\theta$ represent the superposition of two waves. The harmonic form provides a way to combine them into a single, phase-shifted wave. This makes it much easier to find the maximum and minimum values of the expression and to solve equations of the form $a\sin\theta + b\cos\theta = c$ .

To express $a\sin\theta + b\cos\theta$ in the form $R\sin(\theta + \alpha)$ : $a\sin\theta + b\cos\theta \equiv R\sin(\theta + \alpha) = R(\sin\theta\cos\alpha + \cos\theta\sin\alpha)$ Comparing coefficients: $a = R\cos\alpha$ and $b = R\sin\alpha$ From this, we derive: $R = \sqrt{a^2 + b^2}$ and $\tan\alpha = \frac{b}{a}$

Worked examples

See the formulas applied — reveal one step at a time, like the exam.

EXAMPLE 1

Solve the equation $2\cot^2\theta - 5\csc\theta = 1$ for $0^\circ \le \theta \le 360^\circ$ .

1
Use a Pythagorean Identity: Recall the identity $1 + \cot^2\theta \equiv \csc^2\theta$ . We can rearrange this to $\cot^2\theta \equiv \csc^2\theta - 1$ .

EXAMPLE 2

(i) Express $5\cos x - 2\sin x$ in the form $R\cos(x+\alpha)$ , where $R>0$ and $0^\circ < \alpha < 90^\circ$ . (ii) Hence, solve the equation $5\cos x - 2\sin x = 4$ for $0^\circ \le x \le 360^\circ$ .

1
Set up the identity: We want $5\cos x - 2\sin x \equiv R\cos(x+\alpha)$ .

How it all connects

The big idea sits in the middle — tap a linked idea to explore the link.

MAIN IDEATrigonometry

Tap a linked idea to see how it connects back to the main topic — that connection is what examiners reward.

Glossary

Try to recall each definition before you reveal it.

Quick check

Answer in your head first — then tap to check. No pressure.

Revision flashcards

Flip the card. Test yourself before the exam.

QUESTION · 1 / 11

Define $\sec\theta$ , $\csc\theta$ , and $\cot\theta$ .

TAP OR SPACE TO FLIP

ANSWER

$\sec\theta = \frac{1}{\cos\theta}$ , $\csc\theta = \frac{1}{\sin\theta}$ , $\cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta}$ .

TAP OR SPACE TO FLIP BACK

1 / 11 · ← → space

Key takeaways

Review these before you close the topic — retrieval beats re-reading.

✓
Use $\cos(2A) = 2\cos^2 A - 1$ when you want to eliminate other trig functions and form a quadratic in $\cos A$ .
✓
Use $\cos(2A) = 1 - 2\sin^2 A$ when you want to form a quadratic in $\sin A$ .
✓
Use $\sin(2A) = 2\sin A \cos A$ to combine or separate terms, often leading to factorisation.

Practice — then mark it

The whole point: a real Cambridge question, marked mark-by-mark.

Real past paper[0]

Practice Questions

Do it on paper → mark itMARKED MARK-BY-MARK · B1 / M1 / A1 · OFFICIAL SCHEME

Extra simulations & links

PhET, GeoGebra and other curated tools — open in a new tab.

Frequently asked

✓

Checkpoint

One marked question is worth ten re-reads — close the loop before you move on.

Checkpoint

Reading it isn’t knowing it — prove it.

Before you move on: do Practice Questions on paper, snap a photo, and get examiner-style feedback on exactly where you win and lose marks.

Attempt & get marked →Takes about a minute · you’ll land back on this lesson

← PREVIOUS2.2 Logarithmic and exponential functions NEXT →2.4 Differentiation

Trigonometry

BY THE END, YOU CAN…

1Recall and use trigonometric identities, including reciprocal functions ($sec$, $csc$, $cot$) and the Pythagorean identities.
2Use compound and double angle formulae to manipulate and simplify trigonometric expressions.
3Express expressions of the form $a\sin\theta + b\cos\theta$ in the harmonic form $R\sin(\theta \pm \alpha)$ or $R\cos(\theta \pm \alpha)$ .
4Solve trigonometric equations for a specified interval, finding all possible solutions.

In simple terms

A friendly intro before the formal notes — no formulas yet.

Trigonometry Toolkit

Master trigonometry by learning how fundamental identities combine to solve complex equations. We'll build from basic rules to powerful techniques for simplifying and solving.

THINK OF IT LIKE…

1
Start with the fundamental identities $\sin^2\theta + \cos^2\theta = 1$ and $\tan\theta = \sin\theta / \cos\theta$ , which arise from the unit circle.
2
Combine two trigonometric terms into one using the harmonic form $a\sin\theta + b\cos\theta = R\sin(\theta + \alpha)$ , where $R = \sqrt{a^2+b^2}$ .
3
Simplify expressions involving different angles using the double angle formulae, like $\sin(2\theta) = 2\sin\theta\cos\theta$ , derived from compound angle rules.
4
To solve an equation, use identities to simplify, find the principal solution, then use the graph's symmetry to find all other solutions in the required interval.

Explore the concept

Use the live diagram and synced steps — play it or tap a step card to walk through.

Key formulas

Tap any symbol to reveal exactly what it means and its units.

$\sec\theta = \frac{1}{\cos\theta} \quad \csc\theta = \frac{1}{\sin\theta} \quad \cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta}$

Tap a symbol — great for exam definitions

$1 + \tan^2\theta \equiv \sec^2\theta \\ 1 + \cot^2\theta \equiv \csc^2\theta$

Tap a symbol — great for exam definitions

$\sin(2A) \equiv 2\sin A \cos A \\ \cos(2A) \equiv \cos^2 A - \sin^2 A \equiv 2\cos^2 A - 1 \equiv 1 - 2\sin^2 A \\ \tan(2A) \equiv \frac{2\tan A}{1 - \tan^2 A}$

Tap a symbol — great for exam definitions

$a\sin\theta + b\cos\theta$

Tap a symbol — great for exam definitions

Full topic notes

Formal explanation with the rigour you need for the exam.

Reciprocal and Pythagorean Identities

$\sec\theta = \frac{1}{\cos\theta} \quad \csc\theta = \frac{1}{\sin\theta} \quad \cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta}$

$1 + \tan^2\theta \equiv \sec^2\theta \\ 1 + \cot^2\theta \equiv \csc^2\theta$

Double Angle Formulae

$\sin(2A) \equiv 2\sin A \cos A \\ \cos(2A) \equiv \cos^2 A - \sin^2 A \equiv 2\cos^2 A - 1 \equiv 1 - 2\sin^2 A \\ \tan(2A) \equiv \frac{2\tan A}{1 - \tan^2 A}$

Use $\cos(2A) = 2\cos^2 A - 1$ when you want to eliminate other trig functions and form a quadratic in $\cos A$ .
Use $\cos(2A) = 1 - 2\sin^2 A$ when you want to form a quadratic in $\sin A$ .
Use $\sin(2A) = 2\sin A \cos A$ to combine or separate terms, often leading to factorisation.

EXAM TIP

Harmonic Form: $a\sin\theta + b\cos\theta$

Worked examples

See the formulas applied — reveal one step at a time, like the exam.

EXAMPLE 1

Solve the equation $2\cot^2\theta - 5\csc\theta = 1$ for $0^\circ \le \theta \le 360^\circ$ .

1
Use a Pythagorean Identity: Recall the identity $1 + \cot^2\theta \equiv \csc^2\theta$ . We can rearrange this to $\cot^2\theta \equiv \csc^2\theta - 1$ .

EXAMPLE 2

1
Set up the identity: We want $5\cos x - 2\sin x \equiv R\cos(x+\alpha)$ .

How it all connects

The big idea sits in the middle — tap a linked idea to explore the link.

MAIN IDEATrigonometry

Tap a linked idea to see how it connects back to the main topic — that connection is what examiners reward.

Glossary

Try to recall each definition before you reveal it.

Quick check

Answer in your head first — then tap to check. No pressure.

Revision flashcards

Flip the card. Test yourself before the exam.

QUESTION · 1 / 11

Define $\sec\theta$ , $\csc\theta$ , and $\cot\theta$ .

TAP OR SPACE TO FLIP

ANSWER

$\sec\theta = \frac{1}{\cos\theta}$ , $\csc\theta = \frac{1}{\sin\theta}$ , $\cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta}$ .

TAP OR SPACE TO FLIP BACK

1 / 11 · ← → space

Key takeaways

Review these before you close the topic — retrieval beats re-reading.

✓
Use $\cos(2A) = 2\cos^2 A - 1$ when you want to eliminate other trig functions and form a quadratic in $\cos A$ .
✓
Use $\cos(2A) = 1 - 2\sin^2 A$ when you want to form a quadratic in $\sin A$ .
✓
Use $\sin(2A) = 2\sin A \cos A$ to combine or separate terms, often leading to factorisation.

Practice — then mark it

The whole point: a real Cambridge question, marked mark-by-mark.

Real past paper[0]

Practice Questions

Do it on paper → mark itMARKED MARK-BY-MARK · B1 / M1 / A1 · OFFICIAL SCHEME

Extra simulations & links

PhET, GeoGebra and other curated tools — open in a new tab.

Frequently asked

✓

Checkpoint

One marked question is worth ten re-reads — close the loop before you move on.

Checkpoint

Reading it isn’t knowing it — prove it.

Before you move on: do Practice Questions on paper, snap a photo, and get examiner-style feedback on exactly where you win and lose marks.

Attempt & get marked →Takes about a minute · you’ll land back on this lesson

← PREVIOUS2.2 Logarithmic and exponential functions NEXT →2.4 Differentiation

Revise Trigonometry for Cambridge 9709 (2.3)

In simple terms

Explore the concept

Key formulas

Full topic notes

Reciprocal and Pythagorean Identities

Double Angle Formulae

Harmonic Form: $a\sin\theta + b\cos\theta$

Worked examples

How it all connects

Glossary

Quick check

Revision flashcards

Key takeaways

Practice — then mark it

Practice Questions

Extra simulations & links

Frequently asked

Checkpoint

Reading it isn’t knowing it — prove it.

In simple terms

Explore the concept

Key formulas

Full topic notes

Reciprocal and Pythagorean Identities

Double Angle Formulae

Harmonic Form: $a\sin\theta + b\cos\theta$

Worked examples

How it all connects

Glossary

Quick check

Revision flashcards

Key takeaways

Practice — then mark it

Practice Questions

Extra simulations & links

Frequently asked

Checkpoint

Reading it isn’t knowing it — prove it.

Discuss Trigonometry