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.

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3.3Mathematics22 min readPREMIUM

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…

exam checklist — tick these off

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.

STUDY MODE

READING

TYPEFACE

Literata — a reading serif made for e-books; set like a textbook, sits well beside maths.

SIZE

AIRY SPACING

Wider letter and line spacing. The one change with solid evidence for readers who lose their place — worth a try whoever you are.

Pick whatever reads fastest for you — it differs from person to person, by up to a third.

EXPLAIN SIMPLER

The same notes in plain English — no jargon. Key points and exam tips stay.

DIAGRAM NOTES

A small sketch beside each block of notes.

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, PhET or GeoGebra sim, and synced steps — play it, drag controls, or tap a step.

Figure GeoGebra1 / 4

Loading simulation…

Fetching from GeoGebra…

Double angle formulae on a circle

Use the unit circle to verify identity values at key angles.

If the embed does not load, use Open on GeoGebra.

Integral Resources · GeoGebra · GeoGebra Terms of Service

Step-synced diagram — highlights what to look for in the simulation above.

1 more simulation for this topic — run them in the Simulations section below

Simulations

Every simulation here runs the real model — try the steps on a card, then check what you see against the notes.

1 simulation

GeoGebraCore9709 3.3
The R-formula from a diagram
A diagram for a sin θ + b cos θ with sliders for a, b and θ.
Try this
1. Slide a and b and find the length √(a² + b²) on the diagram.
2. Change θ and check that the equation at the top still balances.
3. Swap the sizes of a and b and see what happens to the extra angle.
Look for a sin θ + b cos θ = R sin(θ + α), with R = √(a² + b²) and tan α = b/a: one sine wave, shifted.
Open on GeoGebra
eugeneleefl1 · GeoGebra · GeoGebra Terms of Service

Key formulas

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

KEY RELATION

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

Tap a symbol — great for exam definitions

KEY RELATION

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

Tap a symbol — great for exam definitions

KEY RELATION

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

Tap a symbol — great for exam definitions

KEY RELATION

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

Tap a symbol — great for exam definitions

KEY RELATION

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

Tap a symbol — great for exam definitions

KEY RELATION

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

Tap a symbol — great for exam definitions

KEY RELATION

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

Tap a symbol — great for exam definitions

KEY RELATION

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

Tap a symbol — great for exam definitions

KEY RELATION

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

Tap a symbol — great for exam definitions

KEY RELATION

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

Tap a symbol — great for exam definitions

KEY RELATION

$R\sin(\theta+\alpha)$

Tap a symbol — great for exam definitions

KEY RELATION

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

Tap a symbol — great for exam definitions

KEY RELATION

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

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

Not clear?

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 $\cos A$ .

Not clear?

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}

EXAM TIP

Always check the required range for your solutions (e.g., $0 \le x \le 2\pi$ or $-180^\circ \le x \le 180^\circ$ ) and whether the answer should be in radians or degrees. Forgetting to find all solutions within the range is a very common way to lose marks. Also, when proving an identity, you must start from one side and manipulate it algebraically until it looks identical to the other side. Do not 'solve' it like an equation.

Not clear?

Worked examples

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

reveal slowly — mark each step

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

Key terms for this topic — skim now; the Check step will test them.

Secant: The reciprocal of cosine: $\sec x = \dfrac{1}{\cos x}$ . Undefined wherever $\cos x = 0$ . Related to tangent by the Pythagorean identity $1 + \tan^2 x \equiv \sec^2 x$ .
Cosecant: The reciprocal of sine: $\operatorname{cosec} x = \dfrac{1}{\sin x}$ (also written $\csc x$ ). Undefined wherever $\sin x = 0$ . Related to cotangent by $1 + \cot^2 x \equiv \operatorname{cosec}^2 x$ .
Cotangent: The reciprocal of tangent: $\cot x = \dfrac{1}{\tan x} = \dfrac{\cos x}{\sin x}$ . Undefined wherever $\sin x = 0$ .
Compound angle formulae: The expansions of trigonometric functions of a sum or difference of angles, such as $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$ and $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$ . Setting $B = A$ in them gives the double angle formulae.
Double angle formulae: The expansions of $\sin 2A$ , $\cos 2A$ and $\tan 2A$ obtained by putting $B = A$ in the compound angle formulae: $\sin 2A = 2\sin A \cos A$ , $\cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A$ .
Harmonic form: Writing $a\sin x + b\cos x$ as a single trigonometric function, $R\sin(x + \alpha)$ or $R\cos(x - \alpha)$ , with $R = \sqrt{a^2 + b^2}$ . It makes equations of the form $a\sin x + b\cos x = c$ easy to solve and gives the maximum ( $R$ ) and minimum ( $-R$ ) of the expression directly.

09Remember

Name it

Read the meaning, then pick which of this lesson’s terms it describes. Miss one and you see what your choice really means.

1 of 6

Which of these is being described?

The expansions of trigonometric functions of a sum or difference of angles, such as $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$ and $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$ . Setting $B = A$ in them gives the double angle formulae.

Choose the term this describes.

10Apply

Quick check

Write your answer first, then compare it with the model one — the gap is what you would have lost.

0 of 6 answered

11Understand

Teach it back

If you can explain it simply, you own it — gaps here are marks you’d lose.

Teach it back

Explain this topic as if teaching a friend. We name the gaps an examiner would still dock.

Your explanation — no notes

12Remember

Revision flashcards

Guess first, then flip — retrieval beats re-reading.

Your guess — write it before you look

1 / 17 · ← → space

Key takeaways

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

close the tab only after these stick

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.

Question 1 of 29709/33 · Q10(b)

MathematicsMay/June 2025

(b)

Given that 0 < y < π/2, find the values of y when x = 0.

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

9709/33/M/J/25

Question 2 of 29709/33 · Q5(b)

MathematicsOctober/November 2024

(b)

Solve the equation cos⁴α-sin⁴α = 4 sin²α cos²α for 0° ≤ α ≤ 180°.

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

9709/33/O/N/24

Extra simulations & links

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

Graphs of sec, csc, cot↗

Frequently asked

Should I work in degrees or radians?

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

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

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

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 9709/33 · Q10(b) 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

∫

Discuss Trigonometry

Ask, share and discuss with other Mathematics students

1June 2026 thresholds are live — post your raw marks and we will work out your gap2 #-1Result1 #1Results for AS0 #1One mark off a grade. Is a remark actually worth it?1 #

Create a post Visit s/9709

Revise Trigonometry for Cambridge 9709 (3.3)

Looking for free Mathematics 9709 notes on Trigonometry? This A-Level lesson covers official syllabus point 3.3 on Pure Mathematics 3. Use the visual steps, key terms, and quick checks below — then practise a real 9709 past-paper question with mark-scheme marking. Built for students searching "Trigonometry 9709 revision", "9709 3.3 notes", or a free alternative to paid note sites.

Full Mathematics 9709 course Mark a past paper →9709 subject hub

3.3Mathematics22 min readPREMIUM

Trigonometry

BY THE END, YOU CAN…

exam checklist — tick these off

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.

STUDY MODE

READING

TYPEFACE

Literata — a reading serif made for e-books; set like a textbook, sits well beside maths.

SIZE

AIRY SPACING

Wider letter and line spacing. The one change with solid evidence for readers who lose their place — worth a try whoever you are.

Pick whatever reads fastest for you — it differs from person to person, by up to a third.

EXPLAIN SIMPLER

The same notes in plain English — no jargon. Key points and exam tips stay.

DIAGRAM NOTES

A small sketch beside each block of notes.

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, PhET or GeoGebra sim, and synced steps — play it, drag controls, or tap a step.

Figure GeoGebra1 / 4

Loading simulation…

Fetching from GeoGebra…

Double angle formulae on a circle

Use the unit circle to verify identity values at key angles.

If the embed does not load, use Open on GeoGebra.

Integral Resources · GeoGebra · GeoGebra Terms of Service

Step-synced diagram — highlights what to look for in the simulation above.

1 more simulation for this topic — run them in the Simulations section below

Simulations

Every simulation here runs the real model — try the steps on a card, then check what you see against the notes.

1 simulation

GeoGebraCore9709 3.3
The R-formula from a diagram
A diagram for a sin θ + b cos θ with sliders for a, b and θ.
Try this
1. Slide a and b and find the length √(a² + b²) on the diagram.
2. Change θ and check that the equation at the top still balances.
3. Swap the sizes of a and b and see what happens to the extra angle.
Look for a sin θ + b cos θ = R sin(θ + α), with R = √(a² + b²) and tan α = b/a: one sine wave, shifted.
Open on GeoGebra
eugeneleefl1 · GeoGebra · GeoGebra Terms of Service

Key formulas

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

KEY RELATION

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

Tap a symbol — great for exam definitions

KEY RELATION

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

Tap a symbol — great for exam definitions

KEY RELATION

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

Tap a symbol — great for exam definitions

KEY RELATION

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

Tap a symbol — great for exam definitions

KEY RELATION

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

Tap a symbol — great for exam definitions

KEY RELATION

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

Tap a symbol — great for exam definitions

KEY RELATION

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

Tap a symbol — great for exam definitions

KEY RELATION

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

Tap a symbol — great for exam definitions

KEY RELATION

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

Tap a symbol — great for exam definitions

KEY RELATION

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

Tap a symbol — great for exam definitions

KEY RELATION

$R\sin(\theta+\alpha)$

Tap a symbol — great for exam definitions

KEY RELATION

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

Tap a symbol — great for exam definitions

KEY RELATION

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

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

Not clear?

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 $\cos A$ .

Not clear?

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}

EXAM TIP

Not clear?

Worked examples

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

reveal slowly — mark each step

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

Key terms for this topic — skim now; the Check step will test them.

Secant: The reciprocal of cosine: $\sec x = \dfrac{1}{\cos x}$ . Undefined wherever $\cos x = 0$ . Related to tangent by the Pythagorean identity $1 + \tan^2 x \equiv \sec^2 x$ .
Cosecant: The reciprocal of sine: $\operatorname{cosec} x = \dfrac{1}{\sin x}$ (also written $\csc x$ ). Undefined wherever $\sin x = 0$ . Related to cotangent by $1 + \cot^2 x \equiv \operatorname{cosec}^2 x$ .
Cotangent: The reciprocal of tangent: $\cot x = \dfrac{1}{\tan x} = \dfrac{\cos x}{\sin x}$ . Undefined wherever $\sin x = 0$ .
Compound angle formulae: The expansions of trigonometric functions of a sum or difference of angles, such as $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$ and $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$ . Setting $B = A$ in them gives the double angle formulae.
Double angle formulae: The expansions of $\sin 2A$ , $\cos 2A$ and $\tan 2A$ obtained by putting $B = A$ in the compound angle formulae: $\sin 2A = 2\sin A \cos A$ , $\cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A$ .
Harmonic form: Writing $a\sin x + b\cos x$ as a single trigonometric function, $R\sin(x + \alpha)$ or $R\cos(x - \alpha)$ , with $R = \sqrt{a^2 + b^2}$ . It makes equations of the form $a\sin x + b\cos x = c$ easy to solve and gives the maximum ( $R$ ) and minimum ( $-R$ ) of the expression directly.

09Remember

Name it

Read the meaning, then pick which of this lesson’s terms it describes. Miss one and you see what your choice really means.

1 of 6

Which of these is being described?

Choose the term this describes.

10Apply

Quick check

Write your answer first, then compare it with the model one — the gap is what you would have lost.

0 of 6 answered

11Understand

Teach it back

If you can explain it simply, you own it — gaps here are marks you’d lose.

Teach it back

Explain this topic as if teaching a friend. We name the gaps an examiner would still dock.

Your explanation — no notes

12Remember

Revision flashcards

Guess first, then flip — retrieval beats re-reading.

Your guess — write it before you look

1 / 17 · ← → space

Key takeaways

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

close the tab only after these stick

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.

Question 1 of 29709/33 · Q10(b)

MathematicsMay/June 2025

(b)

Given that 0 < y < π/2, find the values of y when x = 0.

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

9709/33/M/J/25

Question 2 of 29709/33 · Q5(b)

MathematicsOctober/November 2024

(b)

Solve the equation cos⁴α-sin⁴α = 4 sin²α cos²α for 0° ≤ α ≤ 180°.

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

9709/33/O/N/24

Extra simulations & links

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

Graphs of sec, csc, cot↗

Frequently asked

Should I work in degrees or radians?

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

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

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

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 9709/33 · Q10(b) 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

∫

Discuss Trigonometry

Ask, share and discuss with other Mathematics students

1June 2026 thresholds are live — post your raw marks and we will work out your gap2 #-1Result1 #1Results for AS0 #1One mark off a grade. Is a remark actually worth it?1 #

Create a post Visit s/9709

Revise Trigonometry for Cambridge 9709 (3.3)

In simple terms

Explore the concept

Simulations

The R-formula from a diagram

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

Name it

Quick check

Teach it back

Revision flashcards

Key takeaways

Practice — then mark it

Extra simulations & links

Frequently asked

Checkpoint

Reading it isn’t knowing it — prove it.

Discuss Trigonometry

Revise Trigonometry for Cambridge 9709 (3.3)

In simple terms

Explore the concept

Simulations

The R-formula from a diagram

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

Name it

Quick check

Teach it back

Revision flashcards

Key takeaways

Practice — then mark it

Extra simulations & links

Frequently asked

Checkpoint

Reading it isn’t knowing it — prove it.

Discuss Trigonometry