What are the basic trigonometric graphs and their characteristics?

The basic trigonometric graphs include the sine, cosine, and tangent functions. The sine and cosine graphs are periodic with a period of 360 degrees or 2π radians, and they have an amplitude of 1. The sine graph starts at the origin (0,0), while the cosine graph starts at its maximum value (1,0). The tangent graph has a period of 180 degrees or π radians and features vertical asymptotes where the function is undefined. Understanding these characteristics is crucial for Cambridge IGCSE Mathematics students.

How do you determine the amplitude and period of a trigonometric graph?

The amplitude of a trigonometric graph, such as sine or cosine, is the maximum distance from the midline to the peak of the graph. It is determined by the coefficient in front of the sine or cosine function. The period is the length of one complete cycle of the graph and is calculated by dividing 360 degrees (or 2π radians) by the absolute value of the coefficient of the angle variable. For Cambridge IGCSE Mathematics, understanding these concepts helps in graph transformations.

How do phase shifts affect trigonometric graphs?

Phase shifts in trigonometric graphs occur when the graph is horizontally shifted along the x-axis. This shift is determined by the horizontal translation in the function's equation. For example, in y = sin(x - c), the graph is shifted c units to the right. Understanding phase shifts is essential for Cambridge IGCSE students to accurately graph and interpret trigonometric functions.

What is the significance of vertical shifts in trigonometric graphs?

Vertical shifts occur when a trigonometric graph is moved up or down along the y-axis. This is represented by a constant added or subtracted from the function, such as y = sin(x) + d. The graph shifts up by d units if d is positive and down if d is negative. Recognizing vertical shifts is important for Cambridge IGCSE Mathematics students to understand how trigonometric functions can be transformed.

How can trigonometric graphs be used to model real-world phenomena?

Trigonometric graphs are used to model periodic phenomena such as sound waves, tides, and seasonal patterns. By adjusting the amplitude, period, and phase shift, these graphs can represent various cycles and oscillations. Cambridge IGCSE Mathematics students learn to apply these concepts to solve real-world problems, enhancing their understanding of how mathematics describes natural patterns.

Why is "Giving only the principal value when more solutions exist" a common mistake in Trigonometric Graphs?

Why it happens: Calculator only returns one angle. How to avoid it: Always check the range. For = k in [0, 360], there are usually TWO solutions. Source: 0580/42 — recurring

Why is "Using period $360\degree$ for $\tan$" a common mistake in Trigonometric Graphs?

Why it happens: Generalising from and . How to avoid it: has period 180. Add 180 for the next solution, not 360.

Why is "Confusing $\sin(180 - \theta)$ with $\sin(180 + \theta)$" a common mistake in Trigonometric Graphs?

Why it happens: Memorising the wrong identity. How to avoid it: is positive in 1st & 2nd quadrants. So second solution = 180 - principal.

Why is "Plotting $\tan 90\degree$ as a finite value" a common mistake in Trigonometric Graphs?

Why it happens: Calculator shows ERROR, students guess. How to avoid it: 90 is UNDEFINED. The graph has a vertical asymptote there.

TrigonometryCambridge IGCSE Mathematics (0580)

Trigonometric Graphs

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Trigonometric Graphs — Cambridge IGCSE 0580 Maths Extended (2026)

Sine, cosine and tangent graphs from $0\degree$ to $360\degree$ (and beyond). Period, amplitude, key values and where the graphs cross zero — all examinable.

What you’ll learn

Mapped to the Cambridge IGCSE 0580 syllabus (2025-2027).

E8.6 — Recognise, sketch and interpret graphs of $y = \sin x$ , $y = \cos x$ , $y = \tan x$ for $0\degree \le x \le 360\degree$ .

$y = \sin x$

Wave from $-1$ to $1$ , period $360\degree$ . Passes through $(0, 0), (180, 0), (360, 0)$ .

Key features.

Range: $-1 \le y \le 1$ .
Period: $360\degree$ (one full wave from $0\degree$ to $360\degree$ ).
Maximum $y = 1$ at $x = 90\degree$ .
Minimum $y = -1$ at $x = 270\degree$ .
Zeros: $x = 0\degree, 180\degree, 360\degree$ (every $180\degree$ ).

Symmetry.

$\sin(180\degree - x) = \sin x$ (mirror across $90\degree$ ).
$\sin(360\degree - x) = -\sin x$ (mirror across $180\degree$ , then below axis).

Worked. Solve $\sin x = 0.5$ for $0\degree \le x \le 360\degree$ .

Principal value: $x = \sin^{-1}(0.5) = 30\degree$ .
Second solution: $180\degree - 30\degree = 150\degree$ .
Both in range: $x = 30\degree, 150\degree$ .

Range $[-1, 1]$ .
Period $360\degree$ .
Max at $90\degree$ , min at $270\degree$ .
Zeros every $180\degree$ .

$y = \cos x$

Wave from $-1$ to $1$ , period $360\degree$ . Starts at $(0, 1)$ .

Key features.

Range: $-1 \le y \le 1$ .
Period: $360\degree$ .
Maximum $y = 1$ at $x = 0\degree, 360\degree$ .
Minimum $y = -1$ at $x = 180\degree$ .
Zeros: $x = 90\degree, 270\degree$ .

Relationship to sine. $\cos x$ is $\sin x$ shifted left by $90\degree$ : $\cos x = \sin(x + 90\degree).$

Symmetry.

$\cos(360\degree - x) = \cos x$ (even-like, mirror across $0\degree$ and $360\degree$ ).
$\cos(180\degree - x) = -\cos x$ .

Worked. Solve $\cos x = -0.5$ for $0\degree \le x \le 360\degree$ .

$\cos^{-1}(-0.5) = 120\degree$ .
Second solution: $360\degree - 120\degree = 240\degree$ .
$x = 120\degree, 240\degree$ .

Range $[-1, 1]$ .
Period $360\degree$ .
Max at $0\degree$ and $360\degree$ , min at $180\degree$ .
Zeros at $90\degree$ and $270\degree$ .

$y = \tan x$

Period $180\degree$ . Vertical asymptotes at $90\degree$ , $270\degree$ , …

Key features.

Range: all real numbers.
Period: $180\degree$ (NOT $360\degree$ ).
Asymptotes: $x = 90\degree, 270\degree, \ldots$ ( $\cos x = 0$ , so tan undefined).
Zeros: $x = 0\degree, 180\degree, 360\degree$ .
The graph rises from $-\infty$ at just after $90\degree$ (e.g. at $91\degree$ ) to $+\infty$ at just before $90\degree$ (e.g. at $89\degree$ ). Then it repeats.

Worked. Solve $\tan x = 1$ for $0\degree \le x \le 360\degree$ .

$\tan^{-1}(1) = 45\degree$ .
Period $180\degree$ , so next: $45\degree + 180\degree = 225\degree$ .
$x = 45\degree, 225\degree$ .

Range: all reals.
Period $180\degree$ .
Asymptotes at $90\degree, 270\degree$ .
Zeros at $0, 180, 360\degree$ .

Key values to remember

Memorise sin/cos/tan of $0, 30, 45, 60, 90, 180, 270, 360\degree$ .

$x$	$\sin x$	$\cos x$	$\tan x$
$0\degree$	$0$	$1$	$0$
$30\degree$	$\dfrac{1}{2}$	$\dfrac{\sqrt 3}{2}$	$\dfrac{1}{\sqrt 3}$
$45\degree$	$\dfrac{\sqrt 2}{2}$	$\dfrac{\sqrt 2}{2}$	$1$
$60\degree$	$\dfrac{\sqrt 3}{2}$	$\dfrac{1}{2}$	$\sqrt 3$
$90\degree$	$1$	$0$	undef
$180\degree$	$0$	$-1$	$0$
$270\degree$	$-1$	$0$	undef
$360\degree$	$0$	$1$	$0$

Sketch checklist — when asked to sketch a trig graph for $0\degree \le x \le 360\degree$ :

Mark the axes with $90\degree, 180\degree, 270\degree, 360\degree$ on the $x$ -axis.
Mark $-1, 0, 1$ on the $y$ -axis (for sine/cosine).
Plot the maximum, minimum and zero points clearly.
Draw a smooth curve through them.
Label the graph $y = \sin x$ (or whatever).

Memorise $0, 30, 45, 60, 90\degree$ values.
Sketch: mark axes, plot key points, draw smooth curve.
Label the curve.

How it’s examined

Trig graphs appear most years on Paper 2 (2-3 marks: sketch or read a value) and Paper 4 (4-6 marks combined with solving trig equations). Examiner reports flag forgetting the period of tan ( $180\degree$ , not $360\degree$ ) and missing the second solution when solving sine/cosine equations in range.

Sources: Cambridge IGCSE Mathematics 0580 syllabus 2025-2027 (E8.6); 0580/22 May/Jun 2024 — Q12 (trig graph sketch); 0580 Examiner Reports 2022-2024. Last reviewed 2026-05-05.

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Step-by-step worked examples — Trigonometric Graphs

Step-by-step solutions to past-paper-style questions on trigonometric graphs, written exactly the way a tutor would explain them at the board.

1Identify key features of $y = \sin\theta$
Extended• sin graph
Question
State the maximum value, minimum value, and the values of $\theta$ between $0\degree$ and $360\degree$ for which $\sin\theta = 0$ .
Step-by-step solution
1. Step 1
  $\sin\theta$ oscillates between $-1$ and $1$ .
  $\max = 1,\ \min = -1$
2. Step 2
  $\sin\theta = 0$ at $\theta = 0\degree, 180\degree, 360\degree$ .
Answer
Max $1$ , min $-1$ ; zeros at $0\degree, 180\degree, 360\degree$ .
2Use symmetry of $\cos\theta$
Extended• Adapted from 0580/42 May/Jun 2024 Q17• cos graph
Question
$\cos 70\degree = 0.342$ . Use symmetry to write $\cos 290\degree$ .
Step-by-step solution
1. Step 1
  $\cos$ is symmetric about $\theta = 0$ , so $\cos(360\degree - \theta) = \cos\theta$ .
  $\cos 290\degree = \cos 70\degree = 0.342$
Answer
$0.342$
3Solve a basic trig equation using the graph
Extended• solve
Question
Solve $\sin\theta = 0.5$ for $0\degree \le \theta \le 360\degree$ .
Step-by-step solution
1. Step 1
  Principal value.
  $\theta_{1} = \sin^{-1}(0.5) = 30\degree$
2. Step 2
  Second value (sin positive in 1st & 2nd quadrants): $180\degree - 30\degree$ .
  $\theta_{2} = 150\degree$
Answer
$\theta = 30\degree, 150\degree$
4Identify the asymptotes of $\tan\theta$
Extended• tan graph
Question
State the values of $\theta$ between $0\degree$ and $360\degree$ where $\tan\theta$ is undefined.
Step-by-step solution
1. Step 1
  $\tan = \sin/\cos$ , undefined where $\cos = 0$ .
  $\theta = 90\degree,\ 270\degree$
Answer
$90\degree$ and $270\degree$

Key Formulae — Trigonometric Graphs

The formulae you need to memorise for trigonometric graphs on the Cambridge IGCSE 0580 paper, with every variable defined in plain English and a note on when to use it.

$y = \sin\theta$
$\text{Range: } -1 \le y \le 1; \text{ Period: } 360\degree$
When to use
Sketching or solving sine equations.
$y = \cos\theta$
$\text{Range: } -1 \le y \le 1; \text{ Period: } 360\degree$
When to use
Sketching or solving cosine equations.
$y = \tan\theta$
$\text{Range: all reals; Period: } 180\degree;\text{ asymptotes at } 90\degree, 270\degree$
When to use
Sketching or solving tangent equations.
Useful symmetries
$\sin(180\degree - \theta) = \sin\theta;\ \cos(360\degree - \theta) = \cos\theta$
When to use
Find a second solution from a known principal value.

Key Definitions and Keywords — Trigonometric Graphs

Definitions to memorise and the exact keywords mark schemes credit for trigonometric graphs answers — sharpened from recent examiner reports for the 2026 0580 sitting.

Period
Examiner keyword
The smallest positive interval over which the graph repeats.
Amplitude
Examiner keyword
Half the distance between the maximum and minimum values (1 for $\sin$ and $\cos$ ).
Asymptote
Examiner keyword
A line the graph approaches but never touches. $\tan\theta$ has vertical asymptotes where $\cos\theta = 0$ .
Principal value
Examiner keyword
The single value returned by an inverse trig function on a calculator (e.g. $\sin^{-1}$ returns a value between $-90\degree$ and $90\degree$ ).

Common Mistakes and Misconceptions — Trigonometric Graphs

The traps other students keep falling into on trigonometric graphs questions — taken from recent Cambridge IGCSE 0580 examiner reports and mark schemes — and how to avoid them.

✕Giving only the principal value when more solutions exist
0580/42 — recurring
Why it happens
Calculator only returns one angle.
How to avoid it
Always check the range. For $\sin\theta = k$ in $[0\degree, 360\degree]$ , there are usually TWO solutions.
✕Using period $360\degree$ for $\tan$
Why it happens
Generalising from $\sin$ and $\cos$ .
How to avoid it
$\tan$ has period $180\degree$ . Add $180\degree$ for the next solution, not $360\degree$ .
✕Confusing $\sin(180 - \theta)$ with $\sin(180 + \theta)$
Why it happens
Memorising the wrong identity.
How to avoid it
$\sin$ is positive in 1st & 2nd quadrants. So second solution = $180\degree -$ principal.
✕Plotting $\tan 90\degree$ as a finite value
Why it happens
Calculator shows ERROR, students guess.
How to avoid it
$\tan 90\degree$ is UNDEFINED. The graph has a vertical asymptote there.

Practice questions

Exam-style questions with step-by-step worked solutions. Try one before checking the method.

Past paper style quiz

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Video lesson

Short walkthrough of the concepts students most often get stuck on.

Trigonometric Graphs — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

Trigonometric Graphs

Short Study Notes in the form of Slides

Detailed Study Notes

What you’ll learn

How it’s examined

1Identify key features of y=sin⁡θy = \sin\thetay=sinθ

2Use symmetry of cos⁡θ\cos\thetacosθ

3Solve a basic trig equation using the graph

4Identify the asymptotes of tan⁡θ\tan\thetatanθ

y=sin⁡θy = \sin\thetay=sinθ

y=cos⁡θy = \cos\thetay=cosθ

y=tan⁡θy = \tan\thetay=tanθ

Useful symmetries

Period

Amplitude

Asymptote

Principal value

✕Giving only the principal value when more solutions exist

✕Using period 360°360\degree360° for tan⁡\tantan

✕Confusing sin⁡(180−θ)\sin(180 - \theta)sin(180−θ) with sin⁡(180+θ)\sin(180 + \theta)sin(180+θ)

✕Plotting tan⁡90°\tan 90\degreetan90° as a finite value

Practice questions

Past paper style quiz

Assess your understanding

Video lesson

Trigonometric Graphs — frequently asked questions

1Identify key features of $y = \sin\theta$

2Use symmetry of $\cos\theta$

4Identify the asymptotes of $\tan\theta$

$y = \sin\theta$

$y = \cos\theta$

$y = \tan\theta$

✕Using period $360\degree$ for $\tan$

✕Confusing $\sin(180 - \theta)$ with $\sin(180 + \theta)$

✕Plotting $\tan 90\degree$ as a finite value