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Trigonometry

FoundationHigherAQAEdexcelOCR

Practise trigonometry with this free Foundation and Higher GCSE Maths worksheet from Virtus Academy. You'll work through using SOHCAHTOA to find missing sides and angles, building confidence for your exam, and every question comes with worked solutions in the mark scheme. Suitable for AQA, Edexcel and OCR. Use SOHCAHTOA: label the sides, then choose sin, cos or tan.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

Foundation

WorksheetFoundation · PDFMark SchemeFoundation · PDF

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

Trigonometry connects the angles of a right-angled triangle to the ratios of its sides. The three ratios are sine, cosine and tangent, usually remembered as SOHCAHTOA.

Sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent. The hypotenuse is always the longest side, opposite the right angle, and the opposite and adjacent are named relative to the angle you are using.

The first step is always labelling. Mark the hypotenuse, then the side opposite your angle, then the remaining side as adjacent. Once labelled, the two sides involved tell you which ratio to use.

Revision notes

Labelling the sides

The hypotenuse is opposite the right angle. The opposite side faces the angle you are working with, and the adjacent side is next to it.

Relabelling is needed if the question switches to the other angle, since opposite and adjacent swap over.

Choosing the ratio

Look at which two sides are involved — the one you know and the one you want.

Opposite and hypotenuse means sine, adjacent and hypotenuse means cosine, opposite and adjacent means tangent.

Finding an angle

When both sides are known and the angle is missing, use the inverse function on the calculator.

So if \(\sin x = 0.5\), then \(x = \sin^{-1}(0.5) = 30^\circ\). Check the calculator is in degrees mode first.

Key points

  • SOHCAHTOA gives the three ratios.
  • The hypotenuse is opposite the right angle.
  • Opposite faces the angle being used.
  • Label the sides before choosing a ratio.
  • Use the inverse function to find an angle.
  • Check the calculator is in degrees mode.

Worked examples

Example 1

A right-angled triangle has hypotenuse \(10\)cm and an angle of \(30^\circ\). Find the opposite side.

Working

\[\sin 30 = \frac{o}{10}\]opposite and hypotenuse means sine
\[o = 10 \times \sin 30\]rearrange to find the opposite side
\[= 5\text{cm}\]evaluate the answer

Example 2

A triangle has adjacent \(8\)cm and opposite \(6\)cm. Find the angle.

Working

\[\tan x = \frac{6}{8}\]opposite and adjacent means tangent
\[x = \tan^{-1}(0.75)\]use the inverse function
\[= 36.9^\circ\]evaluate to one decimal place

Example 3

A triangle has hypotenuse \(12\)cm and adjacent \(9\)cm. Find the angle.

Working

\[\cos x = \frac{9}{12}\]adjacent and hypotenuse means cosine
\[x = \cos^{-1}(0.75)\]use the inverse function
\[= 41.4^\circ\]evaluate to one decimal place

Common mistakes

  • Mislabelling opposite and adjacent.

    They are named relative to the angle you are using, so relabel if the angle changes.

  • Using the wrong ratio.

    Identify which two sides are involved, then pick the matching ratio.

  • Leaving the calculator in radians.

    Check for the D or DEG indicator before any trigonometry.

  • Forgetting the inverse function when finding an angle.

    sin x = 0.5 needs sin⁻¹, not sin.

Exam tips

  • Label the hypotenuse first, then opposite, then adjacent.
  • Write down which ratio you are using before substituting.
  • Check the calculator is in degrees mode.
  • Give angles to one decimal place unless told otherwise.

Key terms

Hypotenuse
The longest side, opposite the right angle.
Opposite
The side facing the angle being used.
Adjacent
The side next to the angle, not the hypotenuse.
Inverse function
The calculator function used to find an angle from a ratio.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.