# Trigonometric Graphs & Equations

> Mathematics · CIE IGCSE 0580 (2025–2027)
> Source: https://www.owlsprep.com/study/cie-0580-u6-trigonometric-graphs-equations/

This Extended-only guide covers sketching sine, cosine, and tangent graphs for 0° ≤ x ≤ 360°, plus solving trigonometric equations in that range using symmetry or graph inspection, aligned to CIE IGCSE 0580 syllabus E6.4.

**Prerequisites:** [Understanding of sin, cos, tan ratios for right-angled triangles](https://www.owlsprep.com/study/cie-0580-u6-trigonometric-ratios/); [Basic graph sketching skills](https://www.owlsprep.com/study/cie-0580-u2-graph-sketching-basics/)

## Learning objectives

- Recognise and sketch y = sin x, y = cos x, y = tan x for 0° ≤ x ≤ 360°
- Identify key features of each trigonometric graph (roots, peaks, asymptotes)
- Solve trigonometric equations in 0°–360° using graphs or symmetry to find both solutions
- Apply graph properties to answer structured exam-style questions

## Key Features of Standard Trigonometric Graphs

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.

## Sketching Trigonometric Graphs for 0° ≤ x ≤ 360°

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°

**Summary**

- Sin x is a wave starting at (0°, 0), peaking at 90°, crossing zero at 180°, trough at 270°, ending at (360°, 0)
- Cos x is a wave starting at (0°, 1), crossing zero at 90°, trough at 180°, crossing zero at 270°, ending at (360°, 1)
- Tan x has three disconnected curve segments with vertical asymptotes at 90° and 270°

## Solving Trigonometric Equations Using Graph Inspection

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 $\sin 30^\circ = 0.5$, so first solution $x = 30^\circ$
3. Find the second intersection: the sin graph is symmetric above the x-axis between 0° and 180°, so the second solution is $180^\circ - 30^\circ = 150^\circ$
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.

## Finding Second Solutions Using Symmetry Rules

**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 = $180^\circ - \text{principal solution}$
- For $\cos x = k$: Second solution = $360^\circ - \text{principal solution}$
- For $\tan x = k$: Second solution = $180^\circ + \text{principal solution}$

**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°

**Exam command terms**

- **Solve** — Find all valid solutions of the equation in the stated range, which will almost always be 2 solutions for trig equations in 0°–360° *("Solve $\tan x = 2$ for 0° ≤ x ≤ 360°" requires both 63.4° and 243.4° as answers)*

## Common pitfalls

- **Wrong:** Only providing one solution for a trigonometric equation
  - Why it fails: Almost all trig equations in 0°–360° have two valid solutions, and marks are deducted for missing solutions
  - Correct: Always use the symmetry rules or graph inspection to find both solutions, and confirm they are within the required range
- **Wrong:** Drawing tan x as a single continuous wave across 0°–360°
  - Why it fails: Tan x is undefined at 90° and 270°, so it has vertical asymptotes and three separate curve segments
  - Correct: Add dashed vertical asymptotes at 90° and 270°, and draw disconnected curves for each segment between asymptotes
- **Wrong:** Using radian mode on your calculator for calculations
  - Why it fails: CIE IGCSE 0580 exclusively uses degrees for trigonometry, so radian calculations will give incorrect answers
  - Correct: Check your calculator is set to degree mode before starting any trigonometry question, and state all answers in degrees
- **Wrong:** Mixing up the peak positions of sin x and cos x
  - Why it fails: 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: Memorise the key point values: $\sin 0^\circ = 0$, $\cos 0^\circ = 1$ to avoid confusion
- **Wrong:** Including solutions outside the 0° to 360° range
  - Why it fails: Exam questions explicitly limit the range, so solutions outside this range are not accepted even if mathematically valid
  - Correct: After calculating solutions, cross out any values less than 0° or greater than 360°, and only include valid values in your final answer

## 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° | $180^\circ - principal$ | 2 for $-1<k<1$, 1 for $k=\pm1$ |
| $y = \cos x$ | Max=1 at 0°, 360°, Min=-1 at 180°, roots at 90°, 270° | $360^\circ - principal$ | 2 for $-1<k<1$, 1 for $k=\pm1$ |
| $y = \tan x$ | Asymptotes at 90°, 270°, roots at 0°, 180°, 360° | $180^\circ + principal$ | 2 for all real k |

## 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.

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