Solving trigonometry problems Revision Notes for Edexcel GCSE Maths

Solving trigonometry problems

Understanding exact trigonometry values is essential for solving problems without a calculator. These special values appear frequently in GCSE exams, particularly in non-calculator papers.

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Exact trigonometry values are crucial for GCSE success because they allow you to solve trigonometry problems precisely without relying on decimal approximations from a calculator. Mastering these values will significantly improve your problem-solving speed and accuracy.

What are exact trigonometry values?

Exact trigonometry values are the precise sin, cos and tan values for specific angles that can be expressed as fractions or surds, rather than decimal approximations.

You need to memorise these values as they often appear in exam questions where calculators are not permitted. Learning these values will help you solve trigonometry problems quickly and accurately.

Special angle values

The 30-60-90 triangle

This right-angled triangle provides exact values for 30° and 60° angles:

For 30°:

$\sin 30° = \frac{1}{2}$
$\cos 30° = \frac{\sqrt{3}}{2}$
$\tan 30° = \frac{1}{\sqrt{3}}$

For 60°:

$\sin 60° = \frac{\sqrt{3}}{2}$
$\cos 60° = \frac{1}{2}$
$\tan 60° = \sqrt{3}$

The triangle has sides in the ratio 1 : √3 : 2, which helps you remember these relationships.

The 45-45-90 triangle

This isosceles right-angled triangle gives exact values for 45° angles:

$\sin 45° = \frac{1}{\sqrt{2}}$
$\cos 45° = \frac{1}{\sqrt{2}}$
$\tan 45° = 1$

Since this is an isosceles triangle, the two shorter sides are equal, making sin and cos values identical.

Edge case values

You must also know these boundary values:

For 0°:

$\sin 0° = 0$
$\cos 0° = 1$
$\tan 0° = 0$

For 90°:

$\sin 90° = 1$
$\cos 90° = 0$

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tan 90° is undefined - entering this into a calculator gives an error. This is a common source of mistakes in exams, so always remember that tangent is undefined at 90°.

Using exact values in calculations

When solving triangles using exact values, apply the standard trigonometry formulas:

$\cos x° = \frac{\text{adjacent}}{\text{hypotenuse}}$
$\sin x° = \frac{\text{opposite}}{\text{hypotenuse}}$
$\tan x° = \frac{\text{opposite}}{\text{adjacent}}$

Finding side lengths

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Worked Example: Finding Side Length

Triangle ABC has a right angle at B. Angle BAC = 60° and AC = 14 cm. Calculate the length of AB.

Solution: Using $\cos 60° = \frac{\text{adjacent}}{\text{hypotenuse}}$ :

$\cos 60° = \frac{AB}{14}$

$\frac{1}{2} = \frac{AB}{14}$

$AB = 14 × \frac{1}{2} = 7 \text{ cm}$

Answer: AB = 7 cm

Pattern and arrangement problems

Some questions involve arranging identical shapes to create patterns.

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Worked Example: Pattern Arrangement

Amy arranges identical triangles around a point to make a pattern. Each triangle has angles of 30°, 60° and 90°. Show that she can fit exactly 12 triangles around a point.

Solution: Using the triangle: $\sin x° = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{3}{6} = \frac{1}{2}$

Therefore x° = 30°

Since there are 360° around a point: $360° ÷ 30° = 12$

Conclusion: Amy can fit exactly 12 triangles around the point.

Exam tips

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Essential Exam Strategies:

Always sketch the triangle before solving when the problem describes one
For non-calculator questions, you must use exact values - decimal approximations won't earn marks
Write a clear conclusion explaining how your working leads to the answer
Learn these values by heart - they appear regularly in exams

Practice problem

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Practice Challenge:

A vertical flagpole AB is supported with a wire AC at an angle of 60° to the ground. The base of the wire is 2.4 m from the base of the flagpole. Show that the length of the wire AC is 4.8 m.

Hint: Use $\cos 60°$ and the given measurements to find the hypotenuse.

Key takeaways

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Key Points to Remember:

Memorise exact values for 0°, 30°, 45°, 60° and 90°
Use these values in non-calculator questions only
Sketch triangles when problems describe them to help visualise the solution
Remember that tan 90° is undefined and will cause calculator errors
Always show your complete working and write a clear conclusion

Solving trigonometry problems (Edexcel GCSE Maths): Revision Notes