Solving trigonometry problems Revision Notes for AQA GCSE Maths

Solving trigonometry problems

Understanding exact trigonometry values

When solving trigonometry problems, you often need to work with exact values rather than decimal approximations from a calculator. This is particularly important for non-calculator exam questions where you must demonstrate your knowledge of special angles and their precise trigonometric ratios.

Exact trigonometry values are the precise fractional or surd forms of sine, cosine, and tangent for specific angles. These values are essential for GCSE mathematics and must be memorised for exam success.

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In non-calculator questions, you must show your understanding of exact values rather than relying on decimal approximations. This demonstrates deeper mathematical knowledge and is required for full marks.

Special right triangles

Understanding special right triangles is fundamental to mastering exact trigonometric values. These triangles have specific side ratios that make calculations precise and elegant.

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The 30-60-90 triangle

This triangle has sides in the ratio $1 : \sqrt{3} : 2$ . Understanding this ratio helps you remember the exact values:

$\sin 30° = \frac{1}{2}$
$\cos 30° = \frac{\sqrt{3}}{2}$
$\tan 30° = \frac{1}{\sqrt{3}}$
$\sin 60° = \frac{\sqrt{3}}{2}$
$\cos 60° = \frac{1}{2}$
$\tan 60° = \sqrt{3}$

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The 45-45-90 triangle

This isosceles right triangle has sides in the ratio $1 : 1 : \sqrt{2}$ . All the trigonometric values for 45° follow from this relationship:

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

Key exact values to memorise

You must also know the values for 0° and 90° to complete your understanding of exact trigonometric values:

$\sin 0° = 0$ , $\sin 90° = 1$
$\cos 0° = 1$ , $\cos 90° = 0$
$\tan 0° = 0$ , $\tan 90°$ is undefined

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Remember that entering $\tan 90°$ into a calculator will give you an error because this value is undefined. This is a common source of confusion in exams.

Solving problems without a calculator

When you encounter a non-calculator question, follow these systematic steps to ensure accuracy and full marks:

Identify the triangle type - Look for angles of 30°, 45°, 60°, or right angles
Sketch the triangle if not already provided in the question
Label the sides as adjacent, opposite, and hypotenuse relative to the given angle
Choose the correct ratio using SOHCAHTOA
Substitute exact values instead of using decimal approximations
Calculate step by step showing all working clearly

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Worked Example: Finding a missing side

Consider a right triangle where angle $\angle BAC = 60°$ and the hypotenuse $AC = 14$ cm. To find the length of the adjacent side $AB$ :

Step 1: Use the cosine ratio $\cos x° = \frac{\text{adjacent}}{\text{hypotenuse}}$

Step 2: Substitute known values $\cos 60° = \frac{AB}{14}$

Step 3: Use the exact value Since $\cos 60° = \frac{1}{2}$ (exact value)

Step 4: Calculate the result $AB = 14 \times \cos 60° = 14 \times \frac{1}{2} = 7 \text{ cm}$

This demonstrates how using exact values gives you a precise answer without rounding errors.

Pattern recognition in triangles

Some problems involve identical triangles arranged around a point. This type of question tests your understanding that there are 360° around any point.

To solve these problems, determine how many triangles fit by dividing 360° by the angle at the centre of each triangle. For instance, if triangles have a 30° angle at the centre point, then $360° ÷ 30° = 12$ triangles can fit exactly around the point.

When solving these problems, you need to use exact trigonometric values and show that your calculated angle allows the correct number of triangles to fit perfectly around the point.

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Exam Tips for Trigonometry Problems:

Write a clear conclusion explaining how your working leads to the final answer
Show all steps explicitly, especially when substituting exact values
Sketch triangles when they are described in words but not drawn
Check your answer makes sense within the context of the problem
Use exact values throughout your calculations in non-calculator questions
Label diagrams clearly with all given information and what you need to find

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Key Points to Remember:

Learn the exact values for 30°, 45°, 60°, 0°, and 90° - these appear frequently in non-calculator questions
Always sketch triangles when solving problems to visualise the relationships between sides and angles
Use SOHCAHTOA to identify which trigonometric ratio to apply in each situation
Show all working clearly, substituting exact values before calculating the final answer
Remember that $\tan 90°$ is undefined - this will cause an error if entered into a calculator

Solving trigonometry problems (AQA GCSE Maths): Revision Notes