Why can’t x equal 2?

The sine graph only takes values from -1 to 1, so x = 2 has no solutions.

1MA1 · A12, A13AlgebraHigherGrades 6–8

Trigonometric Graphs

Q: What is the period of a graph?

The horizontal length after which it repeats: 360^ for sine and cosine, 180^ for tangent.

Sine, cosine and tangent are defined for any angle, not just those inside a right-angled triangle. Plotting them from $0^{\circ}$ to $360^{\circ}$ produces repeating wave patterns. These graphs explain why $\sin 150^{\circ} = \sin 30^{\circ}$ , and they let you find $\textit{every}$ angle in a range that solves an equation like $\sin x = 0.5$ .

40 min Pythagoras & Trigonometry

The big picture

Each graph repeats: sine and cosine every $360^{\circ}$ , tangent every $180^{\circ}$ . They are also symmetric, so one calculator answer leads to all the others. Recognising the shapes and key points — where each crosses the axis, reaches a maximum or minimum, or has an asymptote — is the main skill.

Edexcel asks you to sketch the graphs, read solutions from them, and apply transformations such as $y = \sin x + 2$ or $y = \cos(x - 30^{\circ})$ , using the same rules as graph transformations.

What you'll learn

Sketch the graphs of $y = \sin x$ , $y = \cos x$ and $y = \tan x$ for $0^{\circ} \le x \le 360^{\circ}$
Identify key features: intercepts, maxima, minima, period and asymptotes
Use symmetry to find all solutions of trigonometric equations in a range
Recognise translations and reflections of trigonometric graphs

The sine and cosine graphs

$y = \sin x$ starts at $(0, 0)$ , rises to a maximum of 1 at $90^{\circ}$ , returns to 0 at $180^{\circ}$ , falls to $-1$ at $270^{\circ}$ and back to 0 at $360^{\circ}$ .

$y = \cos x$ starts at its maximum $(0, 1)$ , crosses zero at $90^{\circ}$ , reaches $-1$ at $180^{\circ}$ , crosses zero at $270^{\circ}$ and returns to 1 at $360^{\circ}$ .

Both have a $\textbf{period}$ of $360^{\circ}$ and values between $-1$ and $1$ . The cosine graph is the sine graph shifted $90^{\circ}$ to the left.

Tip — Mark the key $x$ -values $0^{\circ}, 90^{\circ}, 180^{\circ}, 270^{\circ}, 360^{\circ}$ first; the curve passes through a zero, maximum or minimum at each.

The tangent graph

$y = \tan x$ passes through $(0, 0)$ , $(180^{\circ}, 0)$ and $(360^{\circ}, 0)$ .

It has vertical $\textbf{asymptotes}$ at $x = 90^{\circ}$ and $x = 270^{\circ}$ , where $\cos x = 0$ and the tangent is undefined.

Its period is $180^{\circ}$ and it takes every real value.

Why this works

$\tan x = \dfrac{\sin x}{\cos x}$ , so it is undefined exactly where the cosine graph crosses zero. Seeing the two graphs together explains where the asymptotes must be.

Solving equations

Use the calculator to get one solution, then use the graph’s symmetry and period to find the rest in the range.

For $\sin x = k$ : solutions $x$ and $180^{\circ} - x$ . For $\cos x = k$ : solutions $x$ and $360^{\circ} - x$ . For $\tan x = k$ : solutions $x$ and $x + 180^{\circ}$ .

Sketching the graph with the horizontal line $y = k$ shows how many solutions there are.

Example 1Solve

\cos x = -0.4

for

0^{\circ} \le x \le 360^{\circ}

1Calculator:

\cos^{-1}(-0.4) = 113.6^{\circ}

2Cosine symmetry:

360^{\circ} - 113.6^{\circ} = 246.4^{\circ}

Answer

x = 113.6^{\circ}

246.4^{\circ}

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

Solve $\cos x = -0.7$ for $0^\circ \le x \le 360^\circ$ , to 1 decimal place.

Your answer

Separate with a comma.

Show worked solution

Solution

1 $\cos^{-1}(-0.7) = 134.4^\circ$ .
2By symmetry: $360 - 134.4 = 225.6^\circ$ .

Answer: $x = 134.4^\circ$ or $225.6^\circ$

Example 2Solve

\tan x = 2

for

0^{\circ} \le x \le 360^{\circ}

1Calculator:

\tan^{-1} 2 = 63.4^{\circ}

2Add the period:

63.4^{\circ} + 180^{\circ} = 243.4^{\circ}

Answer

x = 63.4^{\circ}

243.4^{\circ}

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

Solve $\tan x = 0.5$ for $0^\circ \le x \le 360^\circ$ , to 1 decimal place.

Your answer

Separate with a comma.

Show worked solution

Solution

1 $\tan^{-1}(0.5) = 26.6^\circ$ .
2 $\tan$ repeats every $180^\circ$ : $206.6^\circ$ .

Answer: $x = 26.6^\circ$ or $206.6^\circ$

Transformations

$y = \sin x + a$ moves the graph up $a$ ; $y = \sin(x + a)$ moves it left $a$ degrees.

$y = -\cos x$ reflects the graph in the $x$ -axis; $y = \cos(-x)$ reflects in the $y$ -axis — and because the cosine graph is symmetric about the $y$ -axis, it is unchanged.

Track a key point, such as a maximum, to identify a transformation.

Example 1The graph of

y = \sin x

is translated to give a graph with a maximum at

(90^{\circ}, 3)

. Write its equation.

1The original maximum is

(90^{\circ}, 1)

2It has moved up 2, with no horizontal change.

Answer

y = \sin x + 2

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

What is the minimum value of $y = \sin x - 1$ ?

Your answer

Show worked solution

Solution

1The minimum of $\sin x$ is $-1$ .
2Subtracting 1: $-2$ .

Answer: $-2$

Tip — Inside the bracket moves the graph the opposite way to the sign: $\cos(x - 30^{\circ})$ shifts $30^{\circ}$ to the right.

Think like an examiner

Label axes intercepts, maxima, minima and asymptotes on sketches.
Give all solutions within the stated range.
Show the calculator value and how the second solution was found.

Trigonometric graphs

\sin(180^{\circ} - x) = \sin x

Sine symmetry.

\cos(360^{\circ} - x) = \cos x

Cosine symmetry.

\tan(x + 180^{\circ}) = \tan x

Tangent period.

Watch out for these

Giving only the calculator solution.

Use symmetry to find every solution in the range.

Using

180^{\circ} - x

for cosine equations.

For cosine, use

360^{\circ} - x

Drawing

y = \tan x

as a wave between

-1

and 1.

Tangent has asymptotes and takes all values.

Starting the cosine graph at the origin.

\cos 0^{\circ} = 1

Stretch yourself

Solve $2\sin x + 1 = 0$ for $0^{\circ} \le x \le 360^{\circ}$ , using a sketch of $y = \sin x$ to justify your answers.

Hint — Rearrange to $\sin x = -\tfrac12$ . The calculator gives a negative angle, outside the range; use the graph.

Reveal solution

Solution

\sin x = -\tfrac12

. The calculator gives

\sin^{-1}(-\tfrac12) = -30^{\circ}

, which is outside the range. On the sine graph,

\sin x

is negative between

180^{\circ}

and

360^{\circ}

. By symmetry the line

y = -\tfrac12

meets the graph

30^{\circ}

after

180^{\circ}

and

30^{\circ}

before

360^{\circ}

. So

x = 210^{\circ}

x = 330^{\circ}

. Check:

\sin 210^{\circ} = -\tfrac12

✓.

\blacksquare

Questions students ask

What is the period of a graph?

The horizontal length after which it repeats:

360^{\circ}

for sine and cosine,

180^{\circ}

for tangent.

Why can’t

\sin x

equal 2?

The sine graph only takes values from

-1

1

, so

\sin x = 2

has no solutions.

How many solutions does

\sin x = 0.3

have in

0^{\circ}

360^{\circ}

Two — the horizontal line crosses the wave twice in one period.

Key takeaways

Sine and cosine are waves with period $360^{\circ}$ and range $-1$ to $1$ .
Tangent has period $180^{\circ}$ and asymptotes at $90^{\circ}$ and $270^{\circ}$ .
Use symmetry to find all solutions: $180^{\circ} - x$ for sine, $360^{\circ} - x$ for cosine, $+180^{\circ}$ for tangent.
Transformations follow the usual rules for $\mathrm{f}(x) + a$ and $\mathrm{f}(x + a)$ .
Sketch the graph to count and justify solutions.

Exam-style questions

Practise it the way it is examined

3 original questions · 5 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 61 mark

Write down the period of the graph of $y = \cos x$ .[1]
Show mark scheme
- B1 $360^\circ$
Answer: $360^\circ$

Question 2Grade 72 marks

Solve $3\tan x = 3$ for $0^\circ \le x \le 360^\circ$ .[2]
Show mark scheme
- M1 $\tan x = 1$
- A1 $45^\circ$ and $225^\circ$
Answer: $45^\circ$ , $225^\circ$

Question 3Grade 82 marks

Solve $2\cos x + 1 = 0$ for $0^\circ \le x \le 360^\circ$ .[2]
Show mark scheme
- M1 $\cos x = -\frac{1}{2}$
- A1 $120^\circ$ and $240^\circ$
Answer: $120^\circ$ , $240^\circ$

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