Trigonometric Graphs & Equations

Q: Why does tan x have asymptotes at 90° and 270°?

Tan x is equal to . At 90° and 270°, cos x = 0, so dividing by zero is undefined, hence the asymptotes.

Q: How many solutions do trig equations have in 0°–360°?

For sin x = k and cos x = k: 2 solutions if , 1 solution if . For tan x = k: always 2 solutions for any real value of k.

Q: Do I need to use radians for this topic?

No, CIE IGCSE 0580 only uses degrees for trigonometry. Ensure your calculator is set to degree mode for all calculations.

Mathematics· E6.4· 20 min read

1. Key Features of Standard Trigonometric Graphs★★☆☆☆⏱ 5 min

All three basic trigonometric graphs have unique, easily recognisable features in the 0° to 360° range that you need to memorise for sketching and equation solving.

Function	Maximum Value	Minimum Value	Roots (y=0)	Asymptotes
$y = sin x$	1 (at 90°)	-1 (at 270°)	0°, 180°, 360°	None
$y = cos x$	1 (at 0°, 360°)	-1 (at 180°)	90°, 270°	None
$y = tan x$	No upper limit	No lower limit	0°, 180°, 360°	90°, 270°

📐 Worked Example

State the maximum and minimum values of $y = cos x$ , and the x-values (0° ≤ x ≤ 360°) where these occur.

1
Recall the cosine graph starts at its maximum value when x=0°
2
$y_{max} = 1, occurs at x = 0^\circ and x = 360^\circ$
3
The minimum value occurs halfway through the 360° cycle
4
$y_{min} = -1, occurs at x = 180^\circ$

Exam tip:

Always label key points (peaks, troughs, roots, asymptotes) when sketching graphs in the exam, as marks are awarded for these details.

2. Sketching Trigonometric Graphs for 0° ≤ x ≤ 360°★★★☆☆⏱ 5 min

You will be asked to sketch these graphs in both calculator and non-calculator Extended papers, so memorise their shapes and key points to avoid mistakes.

📐 Worked Example

Sketch $y = tan x$ for 0° ≤ x ≤ 360°, labelling all key features.

1
Mark the x-axis from 0° to 360° in 90° increments, and the y-axis from -3 to 3 (tan values grow rapidly near asymptotes so you do not need to extend the axis further)
2
Draw dashed vertical asymptotes at x = 90° and x = 270°, where tan x is undefined
3
Plot key known points: (0°, 0), (45°, 1), (135°, -1), (180°, 0), (225°, 1), (315°, -1), (360°, 0)
4
Draw smooth curves between the points, approaching but never touching the asymptotes, with three separate segments for 0–90°, 90–270°, and 270–360°

3. Solving Trigonometric Equations Using Graph Inspection★★★☆☆⏱ 5 min

Solving a trigonometric equation like $sin x = k$ means finding all x-values in 0°–360° where the graph of the trig function crosses the horizontal line $y = k$ .

📐 Worked Example

Use the graph of $y = sin x$ to solve $sin x = 0.5$ for 0° ≤ x ≤ 360°.

1
Draw the horizontal line $y = 0.5$ on top of the sin x graph
2
Find the first intersection point: you know $sin 3 0^{\circ} = 0.5$ , so first solution $x = 3 0^{\circ}$
3
Find the second intersection: the sin graph is symmetric above the x-axis between 0° and 180°, so the second solution is $18 0^{\circ} - 3 0^{\circ} = 15 0^{\circ}$
4
Confirm both solutions are within 0°–360°, so final solutions are 30° and 150°

Exam tip:

If you are given a printed graph in the exam, use a ruler to draw the horizontal line y=k to make it easier to spot intersection points.

4. Finding Second Solutions Using Symmetry Rules★★★★☆⏱ 5 min

📘 Definition

Trigonometric Solution Symmetry Rules

Simple rules to find the second solution of a trig equation without sketching the full graph, based on the quadrant symmetry of each function.

For $sin x = k$ : Second solution = $18 0^{\circ} - principal solution$
For $cos x = k$ : Second solution = $36 0^{\circ} - principal solution$
For $tan x = k$ : Second solution = $18 0^{\circ} + principal solution$

📐 Worked Example

Given that the principal solution of $cos x = - 0.7$ is 134.4°, find the second solution in 0° ≤ x ≤ 360°.

1
Recall the cosine symmetry rule: second solution = 360° minus principal solution
2
$360^\circ - 134.4^\circ = 225.6^\circ$
3
Verify that $cos 225. 6^{\circ} \approx - 0.7$ , so the second solution is 225.6°

5. Common Pitfalls

Wrong move:

Only providing one solution for a trigonometric equation

Why:

Almost all trig equations in 0°–360° have two valid solutions, and marks are deducted for missing solutions

Correct move:

Always use the symmetry rules or graph inspection to find both solutions, and confirm they are within the required range

Wrong move:

Drawing tan x as a single continuous wave across 0°–360°

Why:

Tan x is undefined at 90° and 270°, so it has vertical asymptotes and three separate curve segments

Correct move:

Add dashed vertical asymptotes at 90° and 270°, and draw disconnected curves for each segment between asymptotes

Wrong move:

Using radian mode on your calculator for calculations

Why:

CIE IGCSE 0580 exclusively uses degrees for trigonometry, so radian calculations will give incorrect answers

Correct move:

Check your calculator is set to degree mode before starting any trigonometry question, and state all answers in degrees

Wrong move:

Mixing up the peak positions of sin x and cos x

Why:

Sin x peaks at 90°, while cos x peaks at 0° and 360°, so mixing these up leads to incorrect sketches and wrong solutions

Correct move:

Memorise the key point values: $sin 0^{\circ} = 0$ , $cos 0^{\circ} = 1$ to avoid confusion

Wrong move:

Including solutions outside the 0° to 360° range

Why:

Exam questions explicitly limit the range, so solutions outside this range are not accepted even if mathematically valid

Correct move:

After calculating solutions, cross out any values less than 0° or greater than 360°, and only include valid values in your final answer

6. Quick Reference Cheatsheet

Function	Key Features 0°–360°	Second Solution Rule	Number of Solutions
$y = sin x$	Max=1 at 90°, Min=-1 at 270°, roots at 0°, 180°, 360°	$18 0^{\circ} - p r in c i p a l$	2 for $- 1 < k < 1$ , 1 for $k = \pm 1$
$y = cos x$	Max=1 at 0°, 360°, Min=-1 at 180°, roots at 90°, 270°	$36 0^{\circ} - p r in c i p a l$	2 for $- 1 < k < 1$ , 1 for $k = \pm 1$
$y = tan x$	Asymptotes at 90°, 270°, roots at 0°, 180°, 360°	$18 0^{\circ} + p r in c i p a l$	2 for all real k

7. Frequently Asked

Why does tan x have asymptotes at 90° and 270°?

Tan x is equal to $\frac{s i n x}{c o s x}$ . At 90° and 270°, cos x = 0, so dividing by zero is undefined, hence the asymptotes.

How many solutions do trig equations have in 0°–360°?

For sin x = k and cos x = k: 2 solutions if $- 1 < k < 1$ , 1 solution if $k = \pm 1$ . For tan x = k: always 2 solutions for any real value of k.

Do I need to use radians for this topic?

No, CIE IGCSE 0580 only uses degrees for trigonometry. Ensure your calculator is set to degree mode for all calculations.

Going deeper

syllabusCIE IGCSE 0580 2025–2027 SyllabusRefer to section E6.4 for this topic

What's Next

Now you have mastered trigonometric graphs and equations for CIE IGCSE 0580 Extended, you are ready to progress to more advanced trigonometry topics tested in Papers 2 and 4. The next core skill is applying trigonometric ratios to 3D shapes, which appears as 4–6 mark structured questions in almost every exam series. You can also practice combining your graph knowledge with right-angled triangle trigonometry and bearings problems, which are common extended-tier question types. Make sure you complete past paper practice for this topic to build speed and accuracy, as trigonometric equations are tested in almost every exam sitting. Always double-check your calculator is in degree mode, and that you have found all required solutions in the stated range before submitting your answer.