Graphs of Trigonometric Functions (Sine, Cosine & Tangent)

Introduction

Trigonometric graphs are fundamental to Pure Mathematics 1 and appear repeatedly throughout the 9709 course. Understanding their shapes, key features, and behaviour for angles of any size — positive, negative, and beyond $360°$ (or $2\pi$ radians) — allows you to solve trigonometric equations graphically, identify the number of solutions in a given interval, and interpret transformations. Examiners regularly test whether candidates can produce accurate sketches annotated with correct coordinates, and whether they understand periodicity and symmetry.

Core Concept

The three core trigonometric functions — sine, cosine, and tangent — are defined for all angles, not merely the acute angles of right-angled triangles. By considering angles on a unit circle, each function generates a characteristic wave (or repeating pattern) as the angle varies continuously.

The Sine Function: $y = \sin x$

Domain: all real numbers ( $x \in \mathbb{R}$ )
Range: $-1 \leq \sin x \leq 1$
Period: $360°$ (or $2\pi$ rad)
The graph passes through the origin, rises to a maximum of $1$ at $x = 90°$ ( $\tfrac{\pi}{2}$ ), returns to $0$ at $x = 180°$ ( $\pi$ ), falls to a minimum of $-1$ at $x = 270°$ ( $\tfrac{3\pi}{2}$ ), and completes one full cycle at $x = 360°$ ( $2\pi$ ).
Symmetry: odd function — $\sin(-x) = -\sin x$ (rotational symmetry about the origin).

The Cosine Function: $y = \cos x$

Domain: all real numbers
Range: $-1 \leq \cos x \leq 1$
Period: $360°$ (or $2\pi$ rad)
The graph starts at a maximum of $1$ when $x = 0°$ , falls to $0$ at $x = 90°$ , reaches a minimum of $-1$ at $x = 180°$ , returns to $0$ at $x = 270°$ , and completes one cycle at $x = 360°$ .
Symmetry: even function — $\cos(-x) = \cos x$ (line symmetry about the $y$ -axis).
Key relationship: $\cos x = \sin(x + 90°)$ ; the cosine graph is the sine graph shifted $90°$ to the left.

The Tangent Function: $y = \tan x$

Domain: all real numbers except $x = 90° + 180°n$ for integer $n$ (i.e. $x \neq \tfrac{\pi}{2} + n\pi$ )
Range: all real numbers ( $y \in \mathbb{R}$ )
Period: $180°$ (or $\pi$ rad) — half that of sine and cosine
The graph passes through the origin, increases without bound as $x \to 90°^-$ , and has vertical asymptotes at $x = \pm 90°, \pm 270°, \ldots$
Symmetry: odd function — $\tan(-x) = -\tan x$ .

Key Formulae & Definitions

\sin(x + 360°) = \sin x \qquad \cos(x + 360°) = \cos x \qquad \tan(x + 180°) = \tan x

\sin(-x) = -\sin x \qquad \cos(-x) = \cos x \qquad \tan(-x) = -\tan x

\sin^2 x + \cos^2 x = 1 \qquad \tan x = \frac{\sin x}{\cos x}

Function	Period	Amplitude	Range	Asymptotes
$y = \sin x$	$360°\ (2\pi)$	$1$	$[-1,\ 1]$	None
$y = \cos x$	$360°\ (2\pi)$	$1$	$[-1,\ 1]$	None
$y = \tan x$	$180°\ (\pi)$	Undefined	$\mathbb{R}$	$x = 90° + 180°n$

Worked Examples

Example 1 — Sketching $y = \sin x$ for $-360° \leq x \leq 360°$ and reading off solutions

Question: Sketch $y = \sin x$ for $-360° \leq x \leq 360°$ and hence state all values of $x$ in this interval for which $\sin x = 0.5$ .

Step 1 — Establish key coordinates.
Mark the $x$ -intercepts: $x = -360°,\ -180°,\ 0°,\ 180°,\ 360°$ .
Mark the maxima: $x = -270°$ and $x = 90°$ , where $\sin x = 1$ .
Mark the minima: $x = -90°$ and $x = 270°$ , where $\sin x = -1$ .

Step 2 — Draw a smooth, continuous wave passing through all these points, with the curve rising from $0$ to $1$ and falling symmetrically.

Step 3 — Draw the horizontal line $y = 0.5$ and identify intersections.
Using the calculator (or known value): $\sin^{-1}(0.5) = 30°$ .
In the range $0°$ to $360°$ : solutions are $x = 30°$ and $x = 180° - 30° = 150°$ .
Extend to negative angles using the odd symmetry $\sin(-x) = -\sin x$ , so in $-360°$ to $0°$ :
$\sin x = 0.5$ at $x = -(360° - 30°) = -330°$ and $x = -(180° + 30°) = -210°$ ...

Let's verify directly: $\sin(-330°) = \sin(30°) = 0.5$ ✓ and $\sin(-210°) = \sin(150°) = 0.5$ ✓

Answer: $x = -330°,\ -210°,\ 30°,\ 150°$

Example 2 — Sketching $y = \cos x$ in radians and counting solutions

Question: Sketch $y = \cos x$ for $0 \leq x \leq 4\pi$ and state the number of solutions of $\cos x = -\tfrac{\sqrt{3}}{2}$ in this interval.

Step 1 — Key coordinates in radians:

$x$	$0$	$\dfrac{\pi}{2}$	$\pi$	$\dfrac{3\pi}{2}$	$2\pi$	$\dfrac{5\pi}{2}$	$3\pi$	$\dfrac{7\pi}{2}$	$4\pi$
$\cos x$	$1$	$0$	$-1$	$0$	$1$	$0$	$-1$	$0$	$1$

Step 2 — Sketch a smooth cosine wave over two full periods $[0,\ 4\pi]$ .

Step 3 — Draw $y = -\tfrac{\sqrt{3}}{2} \approx -0.866$ as a horizontal line.

Step 4 — Identify intersections.
$\cos^{-1}\!\left(-\tfrac{\sqrt{3}}{2}\right) = \tfrac{5\pi}{6}$ .
In $[0,\ 2\pi]$ : solutions are $x = \tfrac{5\pi}{6}$ and $x = 2\pi - \tfrac{5\pi}{6} = \tfrac{7\pi}{6}$ .
In $[2\pi,\ 4\pi]$ (second period): add $2\pi$ to each, giving $x = \tfrac{5\pi}{6} + 2\pi = \tfrac{17\pi}{6}$ and $x = \tfrac{7\pi}{6} + 2\pi = \tfrac{19\pi}{6}$ .

Answer: There are 4 solutions in $[0,\ 4\pi]$ .

Example 3 — Using the tangent graph

Question: By sketching $y = \tan x$ for $-180° < x < 180°$ , state all solutions of $\tan x = 1$ .

Step 1 — Note asymptotes at $x = -90°$ and $x = 90°$ ; the curve passes through $(-180°, 0)$ , $(0°, 0)$ , and increases from $-\infty$ to $+\infty$ in each branch.

Step 2 — Draw $y = 1$ . The line crosses the curve once per period.

Step 3 — First solution: $\tan^{-1}(1) = 45°$ .
Second solution (left branch): $45° - 180° = -135°$ .

Answer: $x = -135°$ and $x = 45°$ .

Common Mistakes & Examiner Pitfalls

Confusing the period of tan with sin/cos. Tangent has period $180°$ ( $\pi$ rad), not $360°$ . Candidates often draw only one solution per $360°$ for tan equations, halving the answer.
Forgetting negative angles. When the interval includes negative values (e.g. $-360° \leq x \leq 360°$ ), the periodic and symmetry properties must be applied in both directions. Always check the full stated interval.
Incorrect range endpoints on sketches. Marks are awarded for correctly labelled intercepts and turning points. Leaving a sketch unlabelled loses method marks.
Asymptotes drawn as solid curves. Asymptotes of $y = \tan x$ must be drawn as dashed vertical lines, not solid ones, and must not be touched by the curve.
Mixing degrees and radians. If the question states radians, use $\pi$ -based labels on the $x$ -axis; if degrees, use degree labels. Mixing the two in one sketch is penalised.
Amplitude errors for $\sin$ and $\cos$ . The range is $[-1, 1]$ ; the graph must never exceed $y = 1$ or go below $y = -1$ . Drawing peaks above $1$ or troughs below $-1$ is a common sketch error.

Practice Questions

Q1. Sketch $y = \sin x$ for $0° \leq x \leq 360°$ , labelling all intercepts and turning points. Hence solve $\sin x = -\tfrac{1}{2}$ for $0° \leq x \leq 360°$ .

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Key points on the sketch: $(0°, 0)$ , $(90°, 1)$ , $(180°, 0)$ , $(270°, -1)$ , $(360°, 0)$ .

$\sin^{-1}\!\left(\tfrac{1}{2}\right) = 30°$ .

Since $\sin x = -\tfrac{1}{2}$ (negative), solutions lie in the third and fourth quadrants:

x = 180° + 30° = 210° \quad \text{and} \quad x = 360° - 30° = 330°

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Q2. Sketch $y = \cos x$ for $-\pi \leq x \leq \pi$ and hence solve $\cos x = \tfrac{1}{2}$ for $-\pi \leq x \leq \pi$ .

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Key points: $(-\pi,\ -1)$ , $\left(-\tfrac{\pi}{2},\ 0\right)$ , $(0,\ 1)$ , $\left(\tfrac{\pi}{2},\ 0\right)$ , $(\pi,\ -1)$ .

$\cos^{-1}\!\left(\tfrac{1}{2}\right) = \tfrac{\pi}{3}$ .

Since cosine is even ( $\cos(-x) = \cos x$ ):

x = \tfrac{\pi}{3} \quad \text{and} \quad x = -\tfrac{\pi}{3}

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Q3. Sketch $y = \tan x$ for $0° < x < 360°$ (excluding the asymptotes), clearly showing the asymptotes. Hence state the number of solutions of $\tan x = -3$ in $0° < x < 360°$ .

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The sketch has two branches: one for $0° < x < 90°$ and one for $90° < x < 270°$ and one for $270° < x < 360°$ .

Asymptotes at $x = 90°$ and $x = 270°$ (dashed vertical lines).

The horizontal line $y = -3$ crosses each of the two branches once (once in $(90°, 270°)$ and once in $(270°, 360°)$ ).

There are 2 solutions in $0° < x < 360°$ .

Using the calculator: $\tan^{-1}(3) \approx 71.6°$ , so the solutions are approximately:

x = 180° - 71.6° = 108.4° \quad \text{and} \quad x = 360° - 71.6° = 288.4°

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Q4. How many solutions does $\cos x = 0.3$ have in the interval $0 \leq x \leq 6\pi$ ? Justify your answer with reference to the graph.

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The interval $[0,\ 6\pi]$ covers 3 complete periods of $y = \cos x$ (each period $= 2\pi$ ).

In each period, the horizontal line $y = 0.3$ (which lies strictly between $0$ and $1$ ) crosses the cosine curve twice (once on the way down from the maximum, once on the way back up).

Total solutions = $3 \times 2 = 6$ .

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Q5. Sketch $y = \sin x$ and $y = \cos x$ on the same axes for $0° \leq x \leq 360°$ . Write down the $x$ -coordinates of the points where the two graphs intersect.

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The graphs intersect where $\sin x = \cos x$ , i.e. $\tan x = 1$ .

x = 45° \quad \text{and} \quad x = 225°

These are the only two intersections in $[0°, 360°]$ . Both intersections occur at $y = \tfrac{\sqrt{2}}{2} \approx 0.707$ .

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Connections

Prerequisite — SOHCAHTOA and exact values: The exact values $\sin 30° = \tfrac{1}{2}$ , $\cos 45° = \tfrac{\sqrt{2}}{2}$ , $\tan 60° = \sqrt{3}$ , etc., are essential for annotating graphs precisely and solving equations analytically rather than only by calculator.
Next — Transformations of trigonometric graphs: Once the parent graphs are mastered, you will apply translations, stretches, and reflections to obtain graphs of $y = a\sin(bx + c) + d$ , reading off new amplitudes, periods, and phase shifts.
Next — Trigonometric identities: The identity $\sin^2 x + \cos^2 x = 1$ and $\tan x = \tfrac{\sin x}{\cos x}$ are read directly from the graphs' properties and become the foundation for simplifying and proving identities.
Next — Solving trigonometric equations: The graphical understanding developed here underpins the algebraic method (principal value + symmetry/periodicity) used to find all solutions in a given interval — a major topic in its own right.

Introduction

Core Concept

The Sine Function: $y = \sin x$

The Cosine Function: $y = \cos x$

The Tangent Function: $y = \tan x$

Key Formulae & Definitions

Worked Examples

Example 1 — Sketching $y = \sin x$ for $-360° \leq x \leq 360°$ and reading off solutions

Example 2 — Sketching $y = \cos x$ in radians and counting solutions

Example 3 — Using the tangent graph

Common Mistakes & Examiner Pitfalls

Practice Questions

Connections

Figures

Keep learning

Introduction

Core Concept

The Sine Function: y=sin⁡xy = \sin xy=sinx

The Cosine Function: y=cos⁡xy = \cos xy=cosx

The Tangent Function: y=tan⁡xy = \tan xy=tanx

Key Formulae & Definitions

Worked Examples

Example 1 — Sketching y=sin⁡xy = \sin xy=sinx for −360°≤x≤360°-360° \leq x \leq 360°−360°≤x≤360° and reading off solutions

Example 2 — Sketching y=cos⁡xy = \cos xy=cosx in radians and counting solutions

Example 3 — Using the tangent graph

Common Mistakes & Examiner Pitfalls

Practice Questions

Connections

Figures

Keep learning

The Sine Function: $y = \sin x$

The Cosine Function: $y = \cos x$

The Tangent Function: $y = \tan x$

Example 1 — Sketching $y = \sin x$ for $-360° \leq x \leq 360°$ and reading off solutions

Example 2 — Sketching $y = \cos x$ in radians and counting solutions