# Trigonometry

> CIE IGCSE Mathematics · CIE IGCSE Maths 0580
> Source: https://www.owlsprep.com/study/cie-0580-u6-overview/
> Weight: 12-15% of total exam marks, across both the non-calculator and calculator structured papers

This unit covers core trigonometric concepts from foundational right-angled triangle calculations to advanced non-right-angled rules, 3D applications, and trigonometric graph interpretation, a high-weight topic for your CIE IGCSE Mathematics exam.

**Prerequisites:** [Basic triangle properties and angle calculation skills](https://www.owlsprep.com/study/cie-0580-u4-geometry-overview/); Proficiency in algebraic rearrangement of formulas

## Learning objectives

- Apply Pythagoras’ theorem and SOHCAHTOA trigonometric ratios to solve 2D right-angled triangle problems including bearings (angles of elevation and depression are Extended only).
- (Extended only) Use the sine rule and cosine rule to calculate unknown sides and angles in any non-right-angled triangle.
- (Extended only) Extend trigonometric problem-solving skills to 3D shapes including cuboids, pyramids and prisms.
- (Extended only) Interpret graphs of sine, cosine and tangent functions, and solve basic trigonometric equations for specified angle ranges between 0° and 360°.

## Unit at a glance

The unit follows a sequential learning arc starting with foundational right-angled trigonometry and Pythagoras’ theorem, before progressing to rules for non-right-angled triangles and 3D problem-solving. The final sub-topic covers trigonometric function graphs and basic equation solving, completing your core trigonometry knowledge for the exam.

You will apply all concepts to common exam scenarios including height and distance calculations, navigation bearings, and geometric shape problems, with a focus on selecting the correct formula for each problem type.

Below are the sub-topics in this unit, ordered by increasing difficulty to build your skills progressively:
- [Pythagoras & Right-Angled Trigonometry](https://www.owlsprep.com/study/cie-0580-u6-pythagoras-right-angled-trigonometry/) — Learn Pythagoras' theorem and SOHCAHTOA ratios to solve right-angled triangle problems including heights, distances and bearings.
- [Sine & Cosine Rules and 3D Trigonometry](https://www.owlsprep.com/study/cie-0580-u6-sine-cosine-rules-and-3d/) — (Extended only) Master sine and cosine rules for non-right-angled triangles, and extend trigonometric skills to solve problems in 3D geometric shapes.
- [Trigonometric Graphs & Equations](https://www.owlsprep.com/study/cie-0580-u6-trigonometric-graphs-equations/) — (Extended only) Interpret sine, cosine and tangent graphs, and solve simple trigonometric equations for specified angle ranges.

## Common pitfalls

- **Wrong:** Using SOHCAHTOA ratios for non-right-angled triangles without splitting them or using the sine/cosine rules first.
  - Why it fails: SOHCAHTOA only applies to triangles with a 90° angle, so using it for other triangle types produces invalid results.
  - Correct: Use the sine or cosine rule for any triangle without a right angle, or split the triangle into two right-angled segments if appropriate.
- **Wrong:** Forgetting that sine is positive in both the first and second quadrants when solving trigonometric equations, leading to missing valid solutions.
  - Why it fails: The sine function is symmetric across 90°, so it returns the same value for θ and 180°-θ in the 0° to 360° range.
  - Correct: Always check for both valid solutions in the given angle range when solving equations involving the sine function.
- **Wrong:** Mixing up side and angle pairings when applying the sine or cosine rule.
  - Why it fails: Both rules rely on matching each side to the angle directly opposite it, so misaligned labels lead to calculation errors.
  - Correct: Label triangles with side $a$ opposite angle $A$, side $b$ opposite angle $B$, etc. before applying either rule.

## Cheatsheet

| Concept | Formula/Key Rule | Use Case | Tier |
| --- | --- | --- | --- |
| Pythagoras' Theorem | $a^2 + b^2 = c^2$ (where $c$ is the hypotenuse) | Find unknown sides of right-angled triangles | Core |
| SOHCAHTOA Ratios | $\sin\theta = \frac{opp}{hyp}, \cos\theta = \frac{adj}{hyp}, \tan\theta = \frac{opp}{adj}$ | Solve right-angled triangle side/angle problems | Core |
| Sine Rule | $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ | Find unknown sides/angles when you know 2 angles + 1 side, or 2 sides + a non-included angle | Extended |
| Cosine Rule | $a^2 = b^2 + c^2 - 2bc\cos A$ | Find unknown sides/angles when you know 2 sides + included angle, or all 3 sides | Extended |
| Trig Graph Periods | sin/cos: 360°, tan: 180° | Identify repeating patterns and solve trigonometric equations for given ranges | Extended |
| 3D Trigonometry | Isolate right-angled triangles inside 3D shapes first | Solve problems involving heights and angles in cuboids, pyramids and prisms | Extended |

## What's next

Start your trigonometry learning journey with the first sub-topic covering Pythagoras' theorem and right-angled trigonometry, which builds the foundational skills you will need for the more advanced sub-topics in this unit. Once you complete all content in this unit, you will move on to the next unit covering statistics and probability.

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