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Exact Trigonometric Values

FoundationHigherAQAEdexcelOCR

Get to grips with exact trigonometric values using these Foundation and Higher GCSE Maths practice questions. The worksheet focuses on recalling exact trigonometric values, and the accompanying mark scheme breaks down each solution clearly. Suitable for AQA, Edexcel and OCR. Download the questions and answers for free. You must recall exact values like sin 30° = ½ and cos 60° = ½.

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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

Some angles have trigonometric values that can be written exactly, without decimals. These are worth memorising because non-calculator papers require them.

The key values are \(\sin 30 = \frac{1}{2}\), \(\cos 60 = \frac{1}{2}\), \(\sin 45 = \cos 45 = \frac{\sqrt{2}}{2}\), and \(\tan 45 = 1\). Also \(\sin 60 = \cos 30 = \frac{\sqrt{3}}{2}\).

At the ends of the range, \(\sin 0 = 0\), \(\cos 0 = 1\), \(\sin 90 = 1\) and \(\cos 90 = 0\). Noticing that sine and cosine swap between \(30^\circ\) and \(60^\circ\) halves the amount you actually need to learn.

Revision notes

The main values

\(\sin 30 = \frac{1}{2}\), \(\sin 45 = \frac{\sqrt{2}}{2}\), \(\sin 60 = \frac{\sqrt{3}}{2}\).

Cosine runs in reverse: \(\cos 30 = \frac{\sqrt{3}}{2}\), \(\cos 45 = \frac{\sqrt{2}}{2}\), \(\cos 60 = \frac{1}{2}\).

Tangent values

\(\tan 30 = \frac{1}{\sqrt{3}}\), \(\tan 45 = 1\) and \(\tan 60 = \sqrt{3}\).

The value \(\tan 45 = 1\) makes sense because at \(45^\circ\) the opposite and adjacent sides are equal.

Where they come from

The \(45^\circ\) values come from a right-angled isosceles triangle with sides \(1\), \(1\) and \(\sqrt{2}\).

The \(30^\circ\) and \(60^\circ\) values come from half an equilateral triangle, with sides \(1\), \(\sqrt{3}\) and \(2\).

Key points

  • \(\sin 30 = \cos 60 = \frac{1}{2}\).
  • \(\sin 45 = \cos 45 = \frac{\sqrt{2}}{2}\).
  • \(\sin 60 = \cos 30 = \frac{\sqrt{3}}{2}\).
  • \(\tan 45 = 1\).
  • \(\sin 0 = 0\) and \(\cos 0 = 1\).
  • Sine and cosine swap between \(30^\circ\) and \(60^\circ\).

Worked examples

Example 1

Write down the exact value of \(\sin 30^\circ\).

Working

\[\text{From the half-equilateral triangle}\]the opposite is 1 and the hypotenuse is 2
\[= \tfrac{1}{2}\]state the exact value

Example 2

Write down the exact value of \(\tan 45^\circ\).

Working

\[\text{Opposite equals adjacent at } 45^\circ\]the triangle is isosceles
\[= 1\]state the exact value

Example 3

Write down the exact value of \(\cos 30^\circ\).

Working

\[\text{Adjacent is } \sqrt{3} \text{, hypotenuse is } 2\]from the half-equilateral triangle
\[= \tfrac{\sqrt{3}}{2}\]state the exact value

Common mistakes

  • Swapping the values for \(30^\circ\) and \(60^\circ\).

    sin 30 is ½ but sin 60 is √3/2. Sine increases as the angle grows.

  • Giving a decimal when an exact value is asked for.

    0.87 is not acceptable when √3/2 is required.

  • Forgetting tan 45 = 1.

    It is the easiest to remember, since the two sides are equal.

  • Mixing up sine and cosine at 0 and 90 degrees.

    sin 0 = 0 and cos 0 = 1, which is the reverse of what many expect.

Exam tips

  • Sketch the two special triangles if you cannot recall a value.
  • Remember sine and cosine swap between 30 and 60 degrees.
  • Leave surds in the answer for exact values.
  • Learn these before a non-calculator paper.

Key terms

Exact value
An answer left as a fraction or surd rather than a decimal.
Surd
A root that cannot be simplified to a whole number.
Isosceles
Having two equal sides.
Equilateral
Having three equal sides.

Related topics

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