A12AlgebraHigherGrades 6–8

Trigonometric Graphs

Q: Why does tan have asymptotes?

x = x x, and x = 0 at 90° and 270°.

In right-angled triangles, angles never exceed 90°. But $sin x$ , $cos x$ and $tan x$ are defined for any angle, and their graphs repeat forever. Knowing these three shapes lets you sketch them, read values, and find every solution of an equation like $sin x = - 0.6$ between 0° and 360°.

45 min AQA GCSE Maths

The big picture

Sine and cosine are waves that repeat every 360° and stay between −1 and 1. Tangent repeats every 180° and has vertical asymptotes where it is undefined.

A calculator gives only one angle for $sin^{- 1}$ , $cos^{- 1}$ or $tan^{- 1}$ . The graph’s symmetry supplies the others — this is the key skill AQA tests.

What you'll learn

Sketch $y = sin x$ , $y = cos x$ and $y = tan x$ for $0^{\circ} \leq x \leq 36 0^{\circ}$
Recall key values and the period of each graph
Use symmetry to solve trigonometric equations in a given interval
Apply translations and reflections to trig graphs

The three graphs

$y = sin x$ : starts at $(0, 0)$ , maximum $(9 0^{\circ}, 1)$ , back to 0 at $18 0^{\circ}$ , minimum $(27 0^{\circ}, - 1)$ , 0 at $36 0^{\circ}$ .

$y = cos x$ : starts at $(0, 1)$ , 0 at $9 0^{\circ}$ , minimum $(18 0^{\circ}, - 1)$ , 0 at $27 0^{\circ}$ , back to 1 at $36 0^{\circ}$ .

$y = tan x$ : passes through $(0, 0)$ , $(18 0^{\circ}, 0)$ and $(36 0^{\circ}, 0)$ , with asymptotes at $x = 9 0^{\circ}$ and $x = 27 0^{\circ}$ .

Why this works

The cosine graph is the sine graph shifted 90° to the left: $cos x = sin (x + 9 0^{\circ})$ . Learning one shape well gives you both.

Tip — Sketch with the axes marked every 90° and label ±1. That is enough for almost every exam question.

Solving equations with symmetry

Sine is symmetrical about $9 0^{\circ}$ (and $27 0^{\circ}$ ): if one solution is $α$ , another is $18 0^{\circ} - α$ .

Cosine is symmetrical about $18 0^{\circ}$ : the second solution is $36 0^{\circ} - α$ .

Tangent repeats every $18 0^{\circ}$ : the second solution is $α + 18 0^{\circ}$ .

x = cos^{- 1} (0.25) = 75. 5^{\circ}

2Other solution:

36 0^{\circ} - 75. 5^{\circ}

Answer

x = 75. 5^{\circ}

x = 284. 5^{\circ}

Your turnSame method, new numbers — try it before moving on.

Solve $cos x = 0.6$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ . Give your answers to 1 decimal place.

Your answer

Separate with a comma, to 1 d.p.

Rearranging first

Make the trig function the subject before using the inverse.

cos x = - 0.5

cos^{- 1} (- 0.5) = 12 0^{\circ}

3Other solution:

36 0^{\circ} - 12 0^{\circ}

Answer

x = 12 0^{\circ}

x = 24 0^{\circ}

Your turnSame method, new numbers — try it before moving on.

Solve $2 sin x - 1 = 0$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ .

Your answer

Separate with a comma.

Tip — Sketch the graph and draw the horizontal line $y = k$ . Counting the crossings tells you how many solutions to find.

Transforming trig graphs

$y = sin x + 2$ moves the wave up 2, so it runs between 1 and 3.

$y = cos (x - 9 0^{\circ})$ moves the cosine graph 90° right, which gives the sine graph.

$y = - sin x$ flips the wave in the $x$ -axis, so its first turning point is a minimum at $(9 0^{\circ}, - 1)$ .

Your turnSame method, new numbers — try it before moving on.

What is the maximum value of $y = sin x + 2$ ?

Your answer

Think like an examiner

Give all solutions in the stated range.
Round angles to 1 decimal place unless told otherwise.
Mark asymptotes on tangent sketches as dashed lines.

Remember these

sin x = sin (18 0^{\circ} - x)

Sine symmetry.

cos x = cos (36 0^{\circ} - x)

Cosine symmetry.

tan x = tan (x + 18 0^{\circ})

Tangent period.

Watch out for these

Giving only the calculator value.

Use the graph to find the second solution.

Using

18 0^{\circ} - α

for cosine.

Cosine uses

36 0^{\circ} - α

Including a negative calculator angle as an answer.

Convert it into the required range.

Drawing

tan x

as a wave between −1 and 1.

It is unbounded with asymptotes at 90° and 270°.

Stretch yourself

Solve $5 sin x + 3 = 1$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ , giving answers to 1 decimal place.

Hint — Rearrange to $sin x = - 0.4$ ; both solutions are between 180° and 360°.

Questions students ask

What is the period of a graph?

The horizontal length of one complete repeat: 360° for sine and cosine, 180° for tangent.

Why does tan have asymptotes?

tan x = \frac{sin x}{cos x}

, and

cos x = 0

at 90° and 270°.

How many solutions does

sin x = 2

have?

None — sine never exceeds 1.

Key takeaways

Sine and cosine: period 360°, range −1 to 1.
Tangent: period 180°, asymptotes at 90° and 270°.
Sine: $18 0^{\circ} - α$ . Cosine: $36 0^{\circ} - α$ . Tangent: $α + 18 0^{\circ}$ .
Rearrange before using inverse functions.
Translations and reflections work as for any graph.

Exam-style questions

Practise it the way it is examined

3 original questions · 7 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 62 marks

(a)Write down the maximum value of $y = 3 cos x$ .[1]
(b)Write down the period of $y = tan x$ .[1]

Question 2Grade 73 marks

Solve $5 sin x + 2 = 0$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ . Give your answers to 1 decimal place.[3]

Question 3Grade 82 marks

Given that $sin 3 0^{\circ} = 0.5$ , use the graph of $y = sin x$ to find all solutions of $sin x = - 0.5$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ .[2]

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Trigonometric Graphs

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Transforming trig graphs

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