What is the difference between a phase shift and a horizontal shift?

They mean the same thing in the context of trigonometric functions. 'Phase shift' is the specific term used for the horizontal translation $c$ of periodic functions like sine and cosine.

How do I decide whether to use a sine or cosine function to model a situation?

It depends on the starting condition at $t=0$ or $x=0$. If the situation starts at a maximum or minimum value, a cosine function ($y=a\cos(bx)+d$) is often easier to use. If it starts at the equilibrium or middle value and is rising, a sine function ($y=a\sin(bx)+d$) is usually more straightforward. Either can model the same situation once you match the amplitude, period, shift and principal axis.

How do I get amplitude, period and principal axis from a real context like tides?

The amplitude $|a|$ is half the difference between the highest and lowest values (e.g. high tide minus low tide, divided by two). The principal axis $d$ is the average of the highest and lowest values. The period is the time for one full cycle (e.g. roughly 12 hours for a tide), and you get $b$ from period $=\frac{2\pi}{b}$.

Why are radians preferred over degrees in IB Maths?

Radians are the natural unit for measuring angles in mathematics, especially in calculus. Key formulae, such as the derivatives of trig functions (the derivative of $\sin x$ is $\cos x$), are only this simple when $x$ is in radians. Modelling questions on Paper 1 almost always use radians, so read $b$ as $\frac{2\pi}{\text{period}}$ in radians.

How does the marking engine treat equivalent forms of an answer?

It accepts equivalent forms. A period written as $\frac{2\pi}{\pi/6}$, $12$ or $12\text{ h}$ all earn the same mark, and a model written as a sine or an equivalent cosine both count. It also applies 'ignore subsequent working' (ISW): once you have written a correct answer, a later tidy-up that does not contradict it will not be penalised.

Trigonometric functions and their graphs —…

Revise Trigonometric functions and their graphs for IB Mathematics: Analysis and Approaches (3.4)

Looking for free IB Mathematics: Analysis and Approaches notes on Trigonometric functions and their graphs? This Higher Level lesson covers official syllabus point 3.4 on Maths AA HL. Use the visual steps, key terms, and quick checks below — then practise with IB criterion marking on Trigonometric functions and their graphs. Built for students searching "IB Maths AA 3.4", "Trigonometric functions and their graphs IB revision", or a free alternative to paid IB note sites.

Trigonometric functions and their graphs

Trigonometric functions are the mathematics of anything that repeats: the swing of a pendulum, alternating current, sound, the daily tide, the yearly temperature, a rotating Ferris wheel. In this lesson we build a secure picture of the three basic graphs, learn to stretch, squash and shift them with the parameters in $y=a\sin(b(x-c))+d$ , and then turn that skill into a modelling tool for real periodic phenomena. As in every calculation topic, the worked examples are laid out exactly the way our engine marks your own practice: the method is on the page, mark by mark.

BY THE END, YOU CAN…

1Sketch and describe the graphs of $y=\sin x$ , $y=\cos x$ and $y=\tan x$ , stating period, range and — for tangent — the vertical asymptotes.
2Interpret $y=a\sin(b(x-c))+d$ : identify the amplitude $|a|$ , period $\tfrac{2\pi}{b}$ , horizontal shift $c$ and vertical shift $d$ .
3Determine $a$ , $b$ , $c$ and $d$ from a given graph or from a described context.
4Model periodic phenomena (tides, temperature, a Ferris wheel) with a sinusoidal function and extract features such as maxima, minima and the times they occur.
5Present working so that method marks (M) and accuracy marks (A) — with follow-through — protect the final answer.

In simple terms

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

Riding the Trigonometric Waves

Trigonometric functions describe repeating patterns, like waves, that appear everywhere from sound to circular motion. We'll explore the basic shapes of these graphs and learn how to stretch, squash, and shift them.

THINK OF IT LIKE…

Imagine you're on a Ferris wheel. As it rotates, your height above the ground goes up and down in a smooth, repeating pattern. If you plot your height over time, you get a sine or cosine wave. This analogy shows how circular motion creates the wave-like graphs we study, with the transformations representing changes to the Ferris wheel's size, speed, or starting position.

1
Begin with the unit circle, a circle with radius 1. As a point moves around it, its y-coordinate traces the sine function and its x-coordinate traces the cosine function.
2
Plot these coordinates against the angle of rotation. This reveals the characteristic wave shapes of $y=\sin x$ and $y=\cos x$ , and the repeating pattern of $y=\tan x$ .
3
Learn to transform these basic graphs. We use the parameters $a, b, c, d$ in $y = a\sin(b(x-c))+d$ to change the amplitude (height), period (width), and position of the wave.
4
Use your knowledge of the transformed graphs to identify key features like maximum and minimum values, and to model real periodic situations such as tides and temperature.

Explore the concept

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

GEOGEBRASTEP 1 / 4 · Step 1

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Trigonometric Function Transformations

Use “Trigonometric Function Transformations” — shift curves or drag points to see equilibrium and slopes change.

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Step 1

Begin with the unit circle, a circle with radius 1. As a point moves around it, its y-coordinate traces the sine function and its x-coordinate traces the cosine function.

Key formulas

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

$y = a\sin(b(x-c)) + d \quad \text{or} \quad y = a\cos(b(x-c)) + d$

Full topic notes

Formal explanation with the rigour you need for the exam.

The three basic graphs: $y=\sin x$, $y=\cos x$, $y=\tan x$

The simplest way to define sine and cosine is with the unit circle: a circle of radius 1 centred at the origin. For a point $P(x,y)$ at angle $\theta$ (in radians) from the positive x-axis, $\cos\theta = x$ and $\sin\theta = y$ . As $P$ travels round the circle, these coordinates oscillate between $-1$ and $1$ , and plotting them against $\theta$ produces the familiar waves. The tangent, $\tan x = \frac{\sin x}{\cos x}$ , behaves very differently: it shoots off to infinity wherever $\cos x = 0$ .

$y=\sin x$ : starts at $(0,0)$ , maximum $1$ at $x=\frac{\pi}{2}$ , back to $0$ at $x=\pi$ , minimum $-1$ at $x=\frac{3\pi}{2}$ , cycle complete at $x=2\pi$ . Range $[-1,1]$ , period $2\pi$ .
$y=\cos x$ : starts at a maximum $(0,1)$ , crosses zero at $x=\frac{\pi}{2}$ , minimum $-1$ at $x=\pi$ , crosses zero at $x=\frac{3\pi}{2}$ , cycle complete at $x=2\pi$ . Range $[-1,1]$ , period $2\pi$ .
$y=\tan x$ : repeating branches with period $\pi$ and range all real numbers. Vertical asymptotes occur where $\cos x = 0$ , i.e. at $x=\frac{\pi}{2}+n\pi$ for integer $n$ . There is no amplitude — the curve is unbounded.
Relationship: $\sin x = \cos\!\big(x-\frac{\pi}{2}\big)$ , so the sine graph is the cosine graph shifted $\frac{\pi}{2}$ to the right.

Transformations: $y=a\sin(b(x-c))+d$

By stretching and shifting the basic graphs we can match a huge range of periodic behaviour. The general sinusoidal form is written below, and each of the four parameters $a$ , $b$ , $c$ , $d$ has a precise geometric meaning.

$y = a\sin(b(x-c)) + d \quad \text{or} \quad y = a\cos(b(x-c)) + d$

$a$ — amplitude. The amplitude is $|a|$ : the distance from the principal axis to a maximum (or minimum). If $a<0$ the graph is also reflected in the principal axis, but the amplitude is still $|a|$ , never negative.
$b$ — period. The period is $\frac{2\pi}{b}$ for sine and cosine (and $\frac{\pi}{b}$ for tangent). A larger $b$ squashes the graph horizontally into more cycles.
$c$ — horizontal shift. The graph moves $c$ units right (left if $c<0$ ). Always factor $b$ out of the bracket first: in $\sin(bx+k)$ rewrite as $\sin\!\big(b(x+\frac{k}{b})\big)$ before reading the shift.
$d$ — vertical shift. The graph moves $d$ units up, so the principal axis is the line $y=d$ . Then maximum $= d+|a|$ and minimum $= d-|a|$ .

EXAM TIP

Check whether the question is in radians or degrees before doing anything — mixing them is a classic slip. And when a phase appears as $\sin(bx+k)$ , always factor out $b$ first to read the horizontal shift as $-\frac{k}{b}$ ; reading it straight off as $k$ is one of the most heavily penalised errors in this topic.

Modelling periodic phenomena: tides, temperature and Ferris wheels

Real repeating situations map straight onto $y=a\sin(b(x-c))+d$ . To build the model, read off four things from the context. The amplitude $|a|$ is half the difference between the highest and lowest values. The principal axis $d$ is their average. The period is the time for one full cycle, which fixes $b=\frac{2\pi}{\text{period}}$ . The horizontal shift $c$ lines the curve up with the starting condition. Once the model is built, features fall out immediately: the maximum is $d+|a|$ , the minimum is $d-|a|$ , and for a sine model starting on the principal axis the first maximum occurs a quarter of a period after the start.

Tides: height oscillates between high and low water; amplitude $=\frac{\text{high}-\text{low}}{2}$ , principal axis $=\frac{\text{high}+\text{low}}{2}$ , period $\approx 12$ h.
Temperature: daily or yearly cycle; amplitude is half the range, principal axis is the mean temperature, period is 24 h or 12 months.
Ferris wheel: height above ground; amplitude $=$ radius, principal axis $=$ radius $+$ boarding height, period $=$ time for one revolution.
Quarter-period rule: for $y=a\sin(b(t-c))+d$ with $a>0$ , the first maximum after the principal-axis crossing at $t=c$ occurs at $t=c+\frac{1}{4}\cdot\frac{2\pi}{b}$ .

Common mistakes examiners penalise

Writing the amplitude as $2a$ or as a negative number — the amplitude is $|a|$ , always positive. For $y=3\sin(2x)+4$ it is $3$ , not $6$ and not $-3$ .
Confusing the coefficient $b$ with the period — the period is $\frac{2\pi}{b}$ , not $b$ . For $y=5\cos\!\big(\frac{\pi}{4}x\big)$ the period is $\frac{2\pi}{\pi/4}=8$ , not $\frac{\pi}{4}$ .
Getting the direction of the horizontal shift wrong — $\sin(x-c)$ shifts RIGHT by $c$ ; $\sin(x+c)$ shifts LEFT. And in $\sin(bx+k)$ you must factor $b$ out first, giving a shift of $-\frac{k}{b}$ , not $k$ .
Taking the maximum as $a$ or the minimum as $-a$ and forgetting $d$ — with the principal axis at $y=d$ , the maximum is $d+|a|$ and the minimum is $d-|a|$ . For $y=4\sin x+7$ the maximum is $11$ , not $4$ .
Reading tangent as if it had an amplitude — $y=\tan x$ is unbounded with period $\pi$ and asymptotes at $x=\frac{\pi}{2}+n\pi$ ; it has no amplitude and no maximum or minimum.
Working in the wrong angle unit — modelling questions are almost always in radians. Check the mode and read $b=\frac{2\pi}{\text{period}}$ in radians.

Model answer — marked the way our engine marks it

On Paper 1 the marks are analytic: each is tied to a specific line of working — a method mark (M) or an accuracy mark (A), where an A-mark depends on the M-mark before it. Follow-through (FT) means a correct step built on an earlier wrong value can still score, and 'ignore subsequent working' (ISW) means a correct answer is not spoiled by a later tidy-up. The engine also accepts equivalent forms. But that protection only exists if the method is written down. Study how each mark below is earned by a specific line.

Where this leads

Reading amplitude, period and shifts fluently is the foundation for solving trigonometric equations graphically, for the double-angle and identity work that follows, and — at HL — for calculus on trigonometric functions, where the derivative of $\sin x$ being $\cos x$ depends on working in radians. The modelling habit you have built here, matching a sinusoid to a real cycle, reappears throughout applied mathematics. Master the routine — read $|a|$ , $\frac{2\pi}{b}$ , $c$ and $d$ ; find maxima as $d+|a|$ ; justify times from the quarter-period; show every line — and periodic modelling becomes variations on a method you already own.

Worked examples

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

EXAMPLE 1

Sketch the graph of the function $f(x) = 3\cos x$ for the domain $0 \le x \le 2\pi$ . Label the coordinates of any maximum points, minimum points and x-intercepts. [4]

1
The function is a vertical stretch of $y=\cos x$ by scale factor 3.

EXAMPLE 2

The graph shows a function of the form $y = a \cos(bx) + d$ . It has a maximum point at A $(0, 5)$ and the next minimum point at B $\left(\frac{\pi}{2}, -1\right)$ . Find the values of $a$ , $b$ and $d$ . [5]

1
1. Vertical shift $d$ — the principal axis is halfway between max and min. $d = \dfrac{\text{max}+\text{min}}{2} = \dfrac{5+(-1)}{2} = 2$ . [M1: averaging max and min] [A1: $d=2$ ]

EXAMPLE 3

The depth of water in a harbour is modelled by $H(t)=2.5\sin\!\big(\tfrac{\pi}{6}t\big)+6$ , where $H$ is the depth in metres and $t$ is the time in hours after midnight. State the amplitude, the period and the principal axis, and find the minimum depth. [4]

1
Compare with $y=a\sin(bt)+d$ : here $a=2.5$ , $b=\frac{\pi}{6}$ , $d=6$ .

EXAMPLE 4

The height of a tide is modelled by $H(t)=4\sin\!\big(\tfrac{\pi}{6}t\big)+7$ , where $H$ is in metres and $t$ is the time in hours after midnight. State the amplitude and the period, and find the maximum height and the first time it occurs. [5]

1
Model answer — full working.

How it all connects

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

MAIN IDEATrigonometric functions and their graphs

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 / 12

What is the definition of $\sin \theta$ and $\cos \theta$ on the unit circle?

TAP OR SPACE TO FLIP

ANSWER

For a point P(x, y) on the unit circle corresponding to an angle $\theta$ from the positive x-axis, $\cos \theta = x$ and $\sin \theta = y$ .

TAP OR SPACE TO FLIP BACK

1 / 12 · ← → space

Key takeaways

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

✓
$y=\sin x$ : starts at $(0,0)$ , maximum $1$ at $x=\frac{\pi}{2}$ , back to $0$ at $x=\pi$ , minimum $-1$ at $x=\frac{3\pi}{2}$ , cycle complete at $x=2\pi$ . Range $[-1,1]$ , period $2\pi$ .
✓
$y=\cos x$ : starts at a maximum $(0,1)$ , crosses zero at $x=\frac{\pi}{2}$ , minimum $-1$ at $x=\pi$ , crosses zero at $x=\frac{3\pi}{2}$ , cycle complete at $x=2\pi$ . Range $[-1,1]$ , period $2\pi$ .
✓
$y=\tan x$ : repeating branches with period $\pi$ and range all real numbers. Vertical asymptotes occur where $\cos x = 0$ , i.e. at $x=\frac{\pi}{2}+n\pi$ for integer $n$ . There is no amplitude — the curve is unbounded.
✓
Relationship: $\sin x = \cos\!\big(x-\frac{\pi}{2}\big)$ , so the sine graph is the cosine graph shifted $\frac{\pi}{2}$ to the right.

Practice — then mark it

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

Real past paper[0]

Practise Graphing and Transformations

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.

Unit Circle and Trig Functions↗

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 Practise Graphing and Transformations 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

← PREVIOUS3.3 The unit circle and radian measure NEXT →3.5 Trigonometric identities and equations

Trigonometric functions and their graphs

BY THE END, YOU CAN…

1Sketch and describe the graphs of $y=\sin x$ , $y=\cos x$ and $y=\tan x$ , stating period, range and — for tangent — the vertical asymptotes.
2Interpret $y=a\sin(b(x-c))+d$ : identify the amplitude $|a|$ , period $\tfrac{2\pi}{b}$ , horizontal shift $c$ and vertical shift $d$ .
3Determine $a$ , $b$ , $c$ and $d$ from a given graph or from a described context.
4Model periodic phenomena (tides, temperature, a Ferris wheel) with a sinusoidal function and extract features such as maxima, minima and the times they occur.
5Present working so that method marks (M) and accuracy marks (A) — with follow-through — protect the final answer.

In simple terms

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

Riding the Trigonometric Waves

THINK OF IT LIKE…

1
Begin with the unit circle, a circle with radius 1. As a point moves around it, its y-coordinate traces the sine function and its x-coordinate traces the cosine function.
2
Plot these coordinates against the angle of rotation. This reveals the characteristic wave shapes of $y=\sin x$ and $y=\cos x$ , and the repeating pattern of $y=\tan x$ .
3
Learn to transform these basic graphs. We use the parameters $a, b, c, d$ in $y = a\sin(b(x-c))+d$ to change the amplitude (height), period (width), and position of the wave.
4
Use your knowledge of the transformed graphs to identify key features like maximum and minimum values, and to model real periodic situations such as tides and temperature.

Explore the concept

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

GEOGEBRASTEP 1 / 4 · Step 1

Loading simulation…

Fetching from GeoGebra…

Trigonometric Function Transformations

Use “Trigonometric Function Transformations” — shift curves or drag points to see equilibrium and slopes change.

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

GeoGebra · GeoGebra Terms of Service

Step 1

Begin with the unit circle, a circle with radius 1. As a point moves around it, its y-coordinate traces the sine function and its x-coordinate traces the cosine function.

Key formulas

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

$y = a\sin(b(x-c)) + d \quad \text{or} \quad y = a\cos(b(x-c)) + d$

Full topic notes

Formal explanation with the rigour you need for the exam.

The three basic graphs: $y=\sin x$, $y=\cos x$, $y=\tan x$

$y=\sin x$ : starts at $(0,0)$ , maximum $1$ at $x=\frac{\pi}{2}$ , back to $0$ at $x=\pi$ , minimum $-1$ at $x=\frac{3\pi}{2}$ , cycle complete at $x=2\pi$ . Range $[-1,1]$ , period $2\pi$ .
$y=\cos x$ : starts at a maximum $(0,1)$ , crosses zero at $x=\frac{\pi}{2}$ , minimum $-1$ at $x=\pi$ , crosses zero at $x=\frac{3\pi}{2}$ , cycle complete at $x=2\pi$ . Range $[-1,1]$ , period $2\pi$ .
$y=\tan x$ : repeating branches with period $\pi$ and range all real numbers. Vertical asymptotes occur where $\cos x = 0$ , i.e. at $x=\frac{\pi}{2}+n\pi$ for integer $n$ . There is no amplitude — the curve is unbounded.
Relationship: $\sin x = \cos\!\big(x-\frac{\pi}{2}\big)$ , so the sine graph is the cosine graph shifted $\frac{\pi}{2}$ to the right.

Transformations: $y=a\sin(b(x-c))+d$

$y = a\sin(b(x-c)) + d \quad \text{or} \quad y = a\cos(b(x-c)) + d$

$a$ — amplitude. The amplitude is $|a|$ : the distance from the principal axis to a maximum (or minimum). If $a<0$ the graph is also reflected in the principal axis, but the amplitude is still $|a|$ , never negative.
$b$ — period. The period is $\frac{2\pi}{b}$ for sine and cosine (and $\frac{\pi}{b}$ for tangent). A larger $b$ squashes the graph horizontally into more cycles.
$c$ — horizontal shift. The graph moves $c$ units right (left if $c<0$ ). Always factor $b$ out of the bracket first: in $\sin(bx+k)$ rewrite as $\sin\!\big(b(x+\frac{k}{b})\big)$ before reading the shift.
$d$ — vertical shift. The graph moves $d$ units up, so the principal axis is the line $y=d$ . Then maximum $= d+|a|$ and minimum $= d-|a|$ .

EXAM TIP

Modelling periodic phenomena: tides, temperature and Ferris wheels

Tides: height oscillates between high and low water; amplitude $=\frac{\text{high}-\text{low}}{2}$ , principal axis $=\frac{\text{high}+\text{low}}{2}$ , period $\approx 12$ h.
Temperature: daily or yearly cycle; amplitude is half the range, principal axis is the mean temperature, period is 24 h or 12 months.
Ferris wheel: height above ground; amplitude $=$ radius, principal axis $=$ radius $+$ boarding height, period $=$ time for one revolution.
Quarter-period rule: for $y=a\sin(b(t-c))+d$ with $a>0$ , the first maximum after the principal-axis crossing at $t=c$ occurs at $t=c+\frac{1}{4}\cdot\frac{2\pi}{b}$ .

Common mistakes examiners penalise

Writing the amplitude as $2a$ or as a negative number — the amplitude is $|a|$ , always positive. For $y=3\sin(2x)+4$ it is $3$ , not $6$ and not $-3$ .
Confusing the coefficient $b$ with the period — the period is $\frac{2\pi}{b}$ , not $b$ . For $y=5\cos\!\big(\frac{\pi}{4}x\big)$ the period is $\frac{2\pi}{\pi/4}=8$ , not $\frac{\pi}{4}$ .
Getting the direction of the horizontal shift wrong — $\sin(x-c)$ shifts RIGHT by $c$ ; $\sin(x+c)$ shifts LEFT. And in $\sin(bx+k)$ you must factor $b$ out first, giving a shift of $-\frac{k}{b}$ , not $k$ .
Taking the maximum as $a$ or the minimum as $-a$ and forgetting $d$ — with the principal axis at $y=d$ , the maximum is $d+|a|$ and the minimum is $d-|a|$ . For $y=4\sin x+7$ the maximum is $11$ , not $4$ .
Reading tangent as if it had an amplitude — $y=\tan x$ is unbounded with period $\pi$ and asymptotes at $x=\frac{\pi}{2}+n\pi$ ; it has no amplitude and no maximum or minimum.
Working in the wrong angle unit — modelling questions are almost always in radians. Check the mode and read $b=\frac{2\pi}{\text{period}}$ in radians.

Model answer — marked the way our engine marks it

Where this leads

Worked examples

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

EXAMPLE 1

Sketch the graph of the function $f(x) = 3\cos x$ for the domain $0 \le x \le 2\pi$ . Label the coordinates of any maximum points, minimum points and x-intercepts. [4]

1
The function is a vertical stretch of $y=\cos x$ by scale factor 3.

EXAMPLE 2

1
1. Vertical shift $d$ — the principal axis is halfway between max and min. $d = \dfrac{\text{max}+\text{min}}{2} = \dfrac{5+(-1)}{2} = 2$ . [M1: averaging max and min] [A1: $d=2$ ]

EXAMPLE 3

1
Compare with $y=a\sin(bt)+d$ : here $a=2.5$ , $b=\frac{\pi}{6}$ , $d=6$ .

EXAMPLE 4

1
Model answer — full working.

How it all connects

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

MAIN IDEATrigonometric functions and their graphs

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 / 12

What is the definition of $\sin \theta$ and $\cos \theta$ on the unit circle?

TAP OR SPACE TO FLIP

ANSWER

For a point P(x, y) on the unit circle corresponding to an angle $\theta$ from the positive x-axis, $\cos \theta = x$ and $\sin \theta = y$ .

TAP OR SPACE TO FLIP BACK

1 / 12 · ← → space

Key takeaways

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

✓
$y=\sin x$ : starts at $(0,0)$ , maximum $1$ at $x=\frac{\pi}{2}$ , back to $0$ at $x=\pi$ , minimum $-1$ at $x=\frac{3\pi}{2}$ , cycle complete at $x=2\pi$ . Range $[-1,1]$ , period $2\pi$ .
✓
$y=\cos x$ : starts at a maximum $(0,1)$ , crosses zero at $x=\frac{\pi}{2}$ , minimum $-1$ at $x=\pi$ , crosses zero at $x=\frac{3\pi}{2}$ , cycle complete at $x=2\pi$ . Range $[-1,1]$ , period $2\pi$ .
✓
$y=\tan x$ : repeating branches with period $\pi$ and range all real numbers. Vertical asymptotes occur where $\cos x = 0$ , i.e. at $x=\frac{\pi}{2}+n\pi$ for integer $n$ . There is no amplitude — the curve is unbounded.
✓
Relationship: $\sin x = \cos\!\big(x-\frac{\pi}{2}\big)$ , so the sine graph is the cosine graph shifted $\frac{\pi}{2}$ to the right.

Practice — then mark it

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

Real past paper[0]

Practise Graphing and Transformations

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.

Unit Circle and Trig Functions↗

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 Practise Graphing and Transformations 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

← PREVIOUS3.3 The unit circle and radian measure NEXT →3.5 Trigonometric identities and equations

Revise Trigonometric functions and their graphs for IB Mathematics: Analysis and Approaches (3.4)

In simple terms

Explore the concept

Key formulas

Full topic notes

The three basic graphs: $y=\sin x$, $y=\cos x$, $y=\tan x$

Transformations: $y=a\sin(b(x-c))+d$

Modelling periodic phenomena: tides, temperature and Ferris wheels

Common mistakes examiners penalise

Model answer — marked the way our engine marks it

Where this leads

Worked examples

How it all connects

Glossary

Quick check

Revision flashcards

Key takeaways

Practice — then mark it

Practise Graphing and Transformations

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 three basic graphs: $y=\sin x$, $y=\cos x$, $y=\tan x$

Transformations: $y=a\sin(b(x-c))+d$

Modelling periodic phenomena: tides, temperature and Ferris wheels

Common mistakes examiners penalise

Model answer — marked the way our engine marks it

Where this leads

Worked examples

How it all connects

Glossary

Quick check

Revision flashcards

Key takeaways

Practice — then mark it

Practise Graphing and Transformations

Extra simulations & links

Frequently asked

Checkpoint

Reading it isn’t knowing it — prove it.

Discuss Trigonometric functions and their graphs