Should I work in degrees or radians?

The question will specify. If it gives a range in degrees, use degrees. If the range is in terms of π, use radians. If no range is given (e.g., in a pure identity proof), it doesn't matter, but radians are standard practice at this level. If the question involves calculus, you MUST use radians.

How do I know which version of the cos(2A) formula to use?

Look at the other terms in the equation. If the equation also contains a 'cos A' term, use the `2cos²A - 1` form to create a quadratic in `cos A`. If it contains a 'sin A' term, use the `1 - 2sin²A` form. If it contains both, you might need a different approach!

What's the point of the harmonic form? Can't I just solve equations without it?

You cannot easily solve an equation like `3sin(x) + 4cos(x) = 2` without the harmonic form. It allows you to reduce two trigonometric terms into one, which is solvable. It's also the only straightforward way to find the maximum or minimum value of such a combination.

I solved for my angle, but my answer was wrong. What did I miss?

There are two common errors. First, your calculator only gives the 'principal value'. You must use this first answer to find all other possible solutions within the given range (e.g., using CAST diagrams or graph sketches). Second, if you squared both sides of an equation at any point, you may have introduced extra solutions that don't work in the original equation, so you must check them.

Where can I learn Trigonometry for Cambridge 9709?

This free lesson on MarkScheme covers syllabus point 3.3 (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/9709/3-3-trigonometry.

Is Trigonometry (3.3) 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 the next level of trigonometry. In P1, you mastered the basics of sine, cosine, and tangent. Now, in P3, we expand our toolkit to handle more complex trigonometric expressions and equations. We will introduce three new functions, powerful identities, and a method for combining sine and cosine waves. These skills are not just for passing exams; they are fundamental in fields like physics, engineering, and signal processing.

BY THE END, YOU CAN…

1Understand and use the definitions and identities of secant, cosecant, and cotangent.
2Apply compound and double angle formulae to prove identities and solve equations.
3Express a sin(x) + b cos(x) in harmonic form to find maximum/minimum values and solve equations.
4Solve trigonometric equations within a given range, paying attention to principal values for inverse functions.

In simple terms

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

Unlocking Trig Equations

P3 trigonometry gives you a powerful toolkit of identities to simplify and solve equations that mix different trig functions. By converting expressions into simpler forms, we can find solutions or determine maximum and minimum values.

THINK OF IT LIKE…

Think of trigonometric identities as a set of universal adaptors. You might have a sine-shaped plug and a cosine-shaped socket. You can't connect them directly. But by using a 'compound angle' or 'harmonic form' adaptor, you can transform one into a shape that fits the other, allowing you to solve the circuit (the equation).

1
Master the core identities, including the Pythagorean set (sin²θ + cos²θ ≡ 1, etc.) and the compound angle formulae. These are your fundamental tools for rewriting expressions.
2
Learn to combine a sin(x) + b cos(x) into a single sine or cosine wave, R sin(x + α). This simplifies finding the amplitude (R) and phase shift (α) of the combined wave.
3
When solving for an angle, like x = arcsin(k), remember that your calculator gives only one answer (the principal value). You must use the unit circle or graphs to find all other solutions in the required range.
4
Always work in radians unless a question specifies degrees. Radians are the natural unit for angles in calculus, which is deeply connected with trigonometry at this level.

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.

$\sec x \equiv \frac{1}{\cos x}$

$1 + \tan^2 x \equiv \sec^2 x$

$\sin(A \pm B) \equiv \sin A \cos B \pm \cos A \sin B$

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.

The Reciprocal Functions: Secant, Cosecant, and Cotangent

We define three new trigonometric functions as the reciprocals of the three you already know. These are secant (sec), cosecant (csc or cosec), and cotangent (cot). Their main purpose is to simplify certain trigonometric expressions and equations, particularly when dealing with fractions.

$\sec x \equiv \frac{1}{\cos x}$

$\csc x \equiv \frac{1}{\sin x}$

$\cot x \equiv \frac{1}{\tan x} \equiv \frac{\cos x}{\sin x}$

Just as $\sin^2 x + \cos^2 x \equiv 1$ is a fundamental identity, we can derive two more 'Pythagorean' identities by dividing this one by $\cos^2 x$ and $\sin^2 x$ respectively. These are essential for solving equations involving the reciprocal functions.

$1 + \tan^2 x \equiv \sec^2 x$

$1 + \cot^2 x \equiv \csc^2 x$

Compound and Double Angle Formulae

These formulae allow us to expand trigonometric functions of sums or differences of angles, like $\sin(A+B)$ . They are provided in the formula book, but you must be fluent in applying them. A particularly important application is deriving the double angle formulae (e.g., for $\sin(2A)$ ) by setting $B=A$ in the compound angle formulae.

$\sin(A \pm B) \equiv \sin A \cos B \pm \cos A \sin B$

$\cos(A \pm B) \equiv \cos A \cos B \mp \sin A \sin B$

$\sin(2A) \equiv 2\sin A \cos A$

$\cos(2A) \equiv \cos^2 A - \sin^2 A \equiv 2\cos^2 A - 1 \equiv 1 - 2\sin^2 A$

Pay close attention to the signs in the compound angle formulae. For $\cos(A+B)$ , the sign on the right-hand side is negative.
The three different forms for $\cos(2A)$ are crucial. Choose the form that best simplifies your equation. For example, if your equation also contains $\cos A$ , use the $2\cos^2 A - 1$ form to create a quadratic in $\cos A$ .

Harmonic Form: Combining Sine and Cosine

An expression of the form $a\sin x + b\cos x$ can be difficult to work with. The harmonic form transforms it into a single trigonometric function, $R\sin(x+\alpha)$ or $R\cos(x-\alpha)$ . This makes it easy to solve equations of the form $a\sin x + b\cos x = c$ and to find the maximum and minimum values of the expression and the angles at which they occur.

To express $a\sin\theta + b\cos\theta$ in the form $R\sin(\theta+\alpha)$ :

$R = \sqrt{a^2+b^2}$

$\tan\alpha = \frac{b}{a}$

Worked examples

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

EXAMPLE 1

Solve the equation $2\tan^2\theta = 3 + \sec\theta$ for $0^\circ \le \theta \le 360^\circ$ .

1
The equation contains both $\tan^2\theta$ and $\sec\theta$ . We need to express it in terms of a single trigonometric function. We use the identity $1 + \tan^2\theta \equiv \sec^2\theta$ , which means $\tan^2\theta \equiv \sec^2\theta - 1$ .

EXAMPLE 2

a) Express $5\sin x - 12\cos x$ in the form $R\sin(x - \alpha)$ , where $R>0$ and $0^\circ < \alpha < 90^\circ$ . State the values of $R$ and $\alpha$ . b) Hence, find the maximum value of the expression and the smallest positive value of $x$ for which it occurs.

1
a) We want to match $5\sin x - 12\cos x$ with $R\sin(x - \alpha) \equiv R(\sin x \cos \alpha - \cos x \sin \alpha)$ . By comparing coefficients of $\sin x$ and $\cos x$ : $R\cos\alpha = 5$ (1) $R\sin\alpha = 12$ (2)

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

What is the definition of sec(x)?

TAP OR SPACE TO FLIP

ANSWER

sec(x) is the secant of x, defined as 1/cos(x). It is undefined when cos(x) = 0.

TAP OR SPACE TO FLIP BACK

1 / 11 · ← → space

Key takeaways

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

✓
Pay close attention to the signs in the compound angle formulae. For $\cos(A+B)$ , the sign on the right-hand side is negative.
✓
The three different forms for $\cos(2A)$ are crucial. Choose the form that best simplifies your equation. For example, if your equation also contains $\cos A$ , use the $2\cos^2 A - 1$ form to create a quadratic in $\cos A$ .

Practice — then mark it

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

Real past paper[0]

Practice P3 Trigonometry 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 P3 Trigonometry 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

← PREVIOUS3.2 Logarithmic and exponential functions NEXT →3.4 Differentiation

Trigonometry

BY THE END, YOU CAN…

1Understand and use the definitions and identities of secant, cosecant, and cotangent.
2Apply compound and double angle formulae to prove identities and solve equations.
3Express a sin(x) + b cos(x) in harmonic form to find maximum/minimum values and solve equations.
4Solve trigonometric equations within a given range, paying attention to principal values for inverse functions.

In simple terms

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

Unlocking Trig Equations

THINK OF IT LIKE…

1
Master the core identities, including the Pythagorean set (sin²θ + cos²θ ≡ 1, etc.) and the compound angle formulae. These are your fundamental tools for rewriting expressions.
2
Learn to combine a sin(x) + b cos(x) into a single sine or cosine wave, R sin(x + α). This simplifies finding the amplitude (R) and phase shift (α) of the combined wave.
3
When solving for an angle, like x = arcsin(k), remember that your calculator gives only one answer (the principal value). You must use the unit circle or graphs to find all other solutions in the required range.
4
Always work in radians unless a question specifies degrees. Radians are the natural unit for angles in calculus, which is deeply connected with trigonometry at this level.

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.

$\sec x \equiv \frac{1}{\cos x}$

$1 + \tan^2 x \equiv \sec^2 x$

$\sin(A \pm B) \equiv \sin A \cos B \pm \cos A \sin B$

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.

The Reciprocal Functions: Secant, Cosecant, and Cotangent

$\sec x \equiv \frac{1}{\cos x}$

$\csc x \equiv \frac{1}{\sin x}$

$\cot x \equiv \frac{1}{\tan x} \equiv \frac{\cos x}{\sin x}$

$1 + \tan^2 x \equiv \sec^2 x$

$1 + \cot^2 x \equiv \csc^2 x$

Compound and Double Angle Formulae

$\sin(A \pm B) \equiv \sin A \cos B \pm \cos A \sin B$

$\cos(A \pm B) \equiv \cos A \cos B \mp \sin A \sin B$

$\sin(2A) \equiv 2\sin A \cos A$

$\cos(2A) \equiv \cos^2 A - \sin^2 A \equiv 2\cos^2 A - 1 \equiv 1 - 2\sin^2 A$

Pay close attention to the signs in the compound angle formulae. For $\cos(A+B)$ , the sign on the right-hand side is negative.
The three different forms for $\cos(2A)$ are crucial. Choose the form that best simplifies your equation. For example, if your equation also contains $\cos A$ , use the $2\cos^2 A - 1$ form to create a quadratic in $\cos A$ .

Harmonic Form: Combining Sine and Cosine

To express $a\sin\theta + b\cos\theta$ in the form $R\sin(\theta+\alpha)$ :

$R = \sqrt{a^2+b^2}$

$\tan\alpha = \frac{b}{a}$

Worked examples

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

EXAMPLE 1

Solve the equation $2\tan^2\theta = 3 + \sec\theta$ for $0^\circ \le \theta \le 360^\circ$ .

1
The equation contains both $\tan^2\theta$ and $\sec\theta$ . We need to express it in terms of a single trigonometric function. We use the identity $1 + \tan^2\theta \equiv \sec^2\theta$ , which means $\tan^2\theta \equiv \sec^2\theta - 1$ .

EXAMPLE 2

1
a) We want to match $5\sin x - 12\cos x$ with $R\sin(x - \alpha) \equiv R(\sin x \cos \alpha - \cos x \sin \alpha)$ . By comparing coefficients of $\sin x$ and $\cos x$ : $R\cos\alpha = 5$ (1) $R\sin\alpha = 12$ (2)

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

What is the definition of sec(x)?

TAP OR SPACE TO FLIP

ANSWER

sec(x) is the secant of x, defined as 1/cos(x). It is undefined when cos(x) = 0.

TAP OR SPACE TO FLIP BACK

1 / 11 · ← → space

Key takeaways

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

✓
Pay close attention to the signs in the compound angle formulae. For $\cos(A+B)$ , the sign on the right-hand side is negative.
✓
The three different forms for $\cos(2A)$ are crucial. Choose the form that best simplifies your equation. For example, if your equation also contains $\cos A$ , use the $2\cos^2 A - 1$ form to create a quadratic in $\cos A$ .

Practice — then mark it

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

Real past paper[0]

Practice P3 Trigonometry 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 P3 Trigonometry 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

← PREVIOUS3.2 Logarithmic and exponential functions NEXT →3.4 Differentiation

Revise Trigonometry for Cambridge 9709 (3.3)

In simple terms

Explore the concept

Key formulas

Full topic notes

The Reciprocal Functions: Secant, Cosecant, and Cotangent

Compound and Double Angle Formulae

Harmonic Form: Combining Sine and Cosine

Worked examples

How it all connects

Glossary

Quick check

Revision flashcards

Key takeaways

Practice — then mark it

Practice P3 Trigonometry 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

The Reciprocal Functions: Secant, Cosecant, and Cotangent

Compound and Double Angle Formulae

Harmonic Form: Combining Sine and Cosine

Worked examples

How it all connects

Glossary

Quick check

Revision flashcards

Key takeaways

Practice — then mark it

Practice P3 Trigonometry Questions

Extra simulations & links

Frequently asked

Checkpoint

Reading it isn’t knowing it — prove it.

Discuss Trigonometry