Trigonometry — Common Values Revision Notes for AQA GCSE Maths

Trigonometry — Common values

Why learn common trigonometric values?

Understanding common trigonometric values is essential for GCSE mathematics success. These special angles appear frequently in exam questions, and knowing their exact values will make calculations much quicker and more accurate. Most importantly, you'll need these values for non-calculator exam sections, so memorising them is crucial.

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Learning these exact values by heart will give you a significant advantage in your GCSE mathematics exam, especially in the non-calculator paper where these values frequently appear.

The two reference triangles

To help you remember the trigonometric values, you can use two special right-angled triangles. These triangles contain the angles 30°, 60°, and 45°, and their side lengths follow predictable patterns.

The 30-60-90 triangle

This triangle has angles of 30°, 60°, and 90°. If the shortest side (opposite the 30° angle) has length 1, then the side opposite the 60° angle has length $\sqrt{3}$ , and the hypotenuse has length 2.

The 45-45-90 triangle

This triangle has two equal angles of 45° and one right angle. If each of the two equal sides has length 1, then the hypotenuse has length $\sqrt{2}$ .

By drawing these triangles and labelling their sides, you can work out all the trigonometric ratios using SOH CAH TOA.

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Master these two reference triangles! The 30-60-90 triangle and the 45-45-90 triangle are the foundation for remembering all common trigonometric values. Practice drawing them from memory until it becomes automatic.

Using SOH CAH TOA with common values

Remember that SOH CAH TOA tells us:

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

Using these ratios with our reference triangles, we can calculate all the common trigonometric values.

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SOH CAH TOA is your key tool This mnemonic is essential for selecting the correct trigonometric ratio. Always identify which sides you know and which you need to find before choosing your ratio.

Common trigonometric values table

Here are the exact values you need to memorise:

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Common Trigonometric Values to Memorise

For 30°, 45°, and 60°:

$\sin 30° = \frac{1}{2}$
$\sin 45° = \frac{1}{\sqrt{2}}$
$\sin 60° = \frac{\sqrt{3}}{2}$
$\cos 30° = \frac{\sqrt{3}}{2}$
$\cos 45° = \frac{1}{\sqrt{2}}$
$\cos 60° = \frac{1}{2}$
$\tan 30° = \frac{1}{\sqrt{3}}$
$\tan 45° = 1$
$\tan 60° = \sqrt{3}$

For 0° and 90°:

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

Note that you cannot work out $\tan 90°$ using triangles - this value is undefined.

Worked example

Let's look at how to apply these common values in practice.

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Worked Example 1: Finding an exact length

Without using a calculator, find the exact length of side $b$ in the right-angled triangle with a 30° angle and hypotenuse of 7 cm.

Solution:

This is a right-angled triangle, so we use SOH CAH TOA to choose the correct trigonometric ratio.
We know the hypotenuse and want to find the adjacent side to the 30° angle, so we use cosine: $\cos 30° = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{b}{7}$
We know that $\cos 30° = \frac{\sqrt{3}}{2}$ , so we substitute this value: $\frac{\sqrt{3}}{2} = \frac{b}{7}$
Solving for $b$ : $b = 7 \times \frac{\sqrt{3}}{2} = \frac{7\sqrt{3}}{2} \text{ cm}$

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Worked Example 2: Proving an identity

Without using a calculator, show that $\cos 30° + \tan 30° = \frac{5\sqrt{3}}{6}$

Solution:

Substitute the known values: $\cos 30° + \tan 30° = \frac{\sqrt{3}}{2} + \frac{1}{\sqrt{3}}$
To add these fractions, we need a common denominator. We can rationalise $\frac{1}{\sqrt{3}}$ : $\frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3}$
Now we have: $\frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{3}$
Using common denominator 6: $\frac{\sqrt{3}}{2} = \frac{3\sqrt{3}}{6} \text{ and } \frac{\sqrt{3}}{3} = \frac{2\sqrt{3}}{6}$
Therefore: $\frac{3\sqrt{3}}{6} + \frac{2\sqrt{3}}{6} = \frac{5\sqrt{3}}{6}$

Practice makes perfect

Try sketching the reference triangles whenever you're unsure about a trigonometric value in an exam. This method provides a quick way to check your work and ensures you're using the correct values.

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Exam Strategy Tip Drawing the reference triangles in your exam can help you verify your trigonometric values and catch any errors before making calculations. This technique is especially valuable in non-calculator questions.

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Key Points to Remember:

Master the two reference triangles (30-60-90 and 45-45-90) to remember all common values
$\sin 30° = \cos 60° = \frac{1}{2}$ , and $\sin 60° = \cos 30° = \frac{\sqrt{3}}{2}$
$\tan 45° = 1$ , which makes sense since the opposite and adjacent sides are equal
For 0° and 90°, remember that $\sin 0° = 0$ and $\cos 0° = 1$ , while $\sin 90° = 1$ and $\cos 90° = 0$
Always rationalise denominators when working with surds in your final answers
Use SOH CAH TOA to select the correct trigonometric ratio for each problem

Trigonometry — Common Values (AQA GCSE Maths): Revision Notes