Teaching unit · UK Year 11 (Key Stage 4 / GCSE Foundation, ages 15 to 16)

Exact trigonometric values

Recalling sin, cos and tan for 0°, 30°, 45°, 60° and 90° without a calculator, and using them for exact answers

About two lessons of 45 to 60 minutes

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Five angles a calculator never needs to be trusted for

For most angles, sin, cos and tan are written as calculator decimals. At the special angles 0°, 30°, 45°, 60° and 90°, the required values can be written exactly using whole numbers, fractions and square roots. Square-root expressions such as $\sqrt{2}$ and $\sqrt{3}$ are called surds. GCSE exams expect these standard exact values to be known, not looked up.

The trick to remembering them is the 45-45-90 and 30-60-90 triangles. A right-angled triangle with two 45° angles has two equal legs; if each leg is 1, Pythagoras gives a hypotenuse of the square root of 2. A right-angled triangle built from an equilateral triangle split in half gives the 30° and 60° values the same way, using the square root of 3.

sin 30° = 1/2 exactlyuse the conventional exact fraction form, and never replace an exact surd with a rounded decimal
A right-angled triangle with legs 1 and 1hypotenuse = $\sqrt{1^{2} + 1^{2}}$ = $\sqrt{2}$ , which is exactly why sin 45° and cos 45° both equal $\sqrt{2}$ /2
A ramp built at exactly 60° to the groundits height-to-length ratio uses sin 60° = $\sqrt{3}$ /2 exactly, with no rounding error
tan 90° is left off the list entirelythere is no exact value, since a right-angled triangle cannot have a second right angle

Learning objective

What students will be able to do

Students will recall the exact values of sin(theta) and cos(theta) for theta = 0, 30, 45, 60, 90 degrees and tan(theta) for theta = 0, 30, 45, 60 degrees without a calculator, use the table in reverse to identify an angle from a given exact value, and apply an exact ratio to find a missing side of a right-angled triangle exactly (as a surd where needed).

Success criteria

I can state the exact value of sin, cos or tan for 0°, 30°, 45°, 60° or 90° from memory, without a calculator.
I can identify which of the five standard angles gives a stated exact sin, cos or tan value.
I can use an exact ratio to find a missing side of a right-angled triangle, giving the answer as an exact value (which may be a surd).

Curriculum anchor

Standards this unit teaches

GCSE Geometry and measures #21UK GCSE Mathematics (DfE, England)
Exact trigonometric values
Subject content statement (Department for Education, "GCSE mathematics: subject content and assessment objectives", published 1 November 2013, reference DFE-00233-2013, "Geometry and measures" section, "Mensuration and calculation", item 21, https://www.gov.uk/government/publications/gcse-mathematics-subject-content-and-assessment-objectives): students should "know the exact values of sin(theta) and cos(theta) for theta = 0, 30, 45, 60 and 90 degrees; know the exact value of tan(theta) for theta = 0, 30, 45 and 60 degrees". This item is entirely underlined type: Foundation content, assessed for every GCSE student (tan 90 degrees is undefined and is correctly not in this list).

Before you start

Prior knowledge

This unit builds on skills students should already have met. Revisit any that are shaky first.

Key vocabulary

Words to teach and display

Exact value: a value given precisely (as a whole number, fraction, or surd), never rounded, so it can never be slightly wrong
Surd: an expression containing an un-simplified square root of a non-square number, such as $\sqrt{2}$ or $\sqrt{3}$ , which cannot be written as a terminating or exact fraction

Teaching sequence

Teach it: concrete, pictorial, abstract

The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.

1. The exact values, and where they come from

Concrete

The non-degenerate 30°, 45° and 60° values can be derived from two special right-angled triangles. The 45° values come from an isosceles right-angled triangle; the 30° and 60° values come from an equilateral triangle split exactly in half. The endpoint values at 0° and 90° follow from the unit-circle definitions (or limiting positions of a right triangle).

A right-angled triangle with both legs equal to 1 has hypotenuse $\sqrt{1^{2} + 1^{2}}$ = $\sqrt{2}$ . This gives sin 45° = opposite/hypotenuse = 1/ $\sqrt{2}$ , which is written with a rational denominator as $\sqrt{2}$ /2. Since the triangle is isosceles, cos 45° is exactly the same value.

theta = 0: sin 0 = 0, cos 0 = 1, tan 0 = 0. theta = 30: sin 30 = 1/2, cos 30 = $\sqrt{3}$ /2, tan 30 = $\sqrt{3}$ /3. theta = 45: sin 45 = cos 45 = $\sqrt{2}$ /2, tan 45 = 1. theta = 60: sin 60 = $\sqrt{3}$ /2, cos 60 = 1/2, tan 60 = $\sqrt{3}$ . theta = 90: sin 90 = 1, cos 90 = 0, tan 90 is undefined.

Worked example

State the exact values of sin 60° and cos 60°, and explain how they relate to sin 30° and cos 30°.

sin 60° = $\sqrt{3}$ /2 and cos 60° = 1/2.
sin 30° = 1/2 and cos 30° = $\sqrt{3}$ /2 — exactly the SWAPPED pair.
This is because 30° and 60° are complementary (they add to 90°), and sin of an angle always equals cos of its complement.

Answer: sin 60° = $\sqrt{3}$ /2, cos 60° = 1/2 (the swap of sin 30° and cos 30°).

Check for understanding, ask

Why do sin 45° and cos 45° come out exactly equal?
Why is there no exact value listed for tan 90°?

2. Using exact values to find a missing side

Abstract

Once the exact ratio for an angle is known, it can be used exactly like any other trigonometric ratio to find a missing side, except the final answer is left as an exact value (a whole number, fraction, or surd) instead of a rounded decimal.

A right-angled triangle has hypotenuse 8 cm and one angle of 30°. Find the side opposite the 30° angle: opposite = hypotenuse x sin 30° = 8 x 1/2 = 4 cm, exactly. Find the side adjacent to the 30° angle: adjacent = hypotenuse x cos 30° = 8 x $\sqrt{3}$ /2 = 4sqrt(3) cm, exactly.

Worked example

A right-angled triangle has an angle of 60° and an adjacent side of 5 cm. Find the side opposite the 60° angle, exactly.

Using tan 60° = $\sqrt{3}$ : opposite = adjacent x tan 60°.
opposite = 5 x $\sqrt{3}$ = 5sqrt(3) cm.

Answer: 5sqrt(3) cm

Check for understanding, ask

Why does the answer 4sqrt(3) cm need to stay as a surd, rather than being rounded to a decimal, in this unit?
Could you find the SAME missing sides using Pythagoras' theorem instead of an exact trig ratio, once you knew both other sides?

Watch for

Common misconceptions and how to address them

Misconception $\sqrt{3}$ /2 as a decimal (about 0.87) is 'more correct' or 'more useful' than leaving it as an exact surd.

Why it happens: Students are used to decimal answers from calculator-based trigonometry and assume a decimal is always the goal.

How to address it: An EXACT answer is a stronger, more precise answer than a rounded decimal: $\sqrt{3}$ /2 is the TRUE value, with zero rounding error, while 0.87 has already lost precision. GCSE mark schemes for 'exact value' questions only accept the exact surd form, not a decimal.

Misconceptiontan 30° = $\sqrt{3}$ , the same as tan 60°.

Why it happens: Students confuse tan 30° and tan 60°, since both involve $\sqrt{3}$ somewhere in the exact-values table.

How to address it: tan 30° = $\sqrt{3}$ /3 (equivalently 1/ $\sqrt{3}$ ), a value LESS than 1 (since 30° is a small angle, so opposite is smaller than adjacent). tan 60° = $\sqrt{3}$ exactly, a value MORE than 1 (60° is a larger angle). Checking whether the angle is under or over 45° (where tan = 1) is a quick sanity check.

Do it together

Guided practice (with answers)

1. State the exact value of sin 30°.
Answer: 1/2
2. State the exact value of cos 45°.
Answer: $\sqrt{2}$ /2
3. State the exact value of tan 60°.
Answer: $\sqrt{3}$
4. For 0° ≤ theta ≤ 90°, find theta for which cos theta = 1/2.
Answer: theta = 60°
5. A right-angled triangle has hypotenuse 6 cm and an angle of 45°. Find the side opposite the 45° angle, exactly.
Answer: 3sqrt(2) cm, because opposite = 6 x sin 45° = 6 x $\sqrt{2}$ /2 = 3sqrt(2).

On their own

Independent practice worksheets

Practise recalling and applying the exact trigonometric values, with computed, never-wrong answer keys.

Exact Trigonometric Values (Year 11)recall the table, and use it in reverse to identify an angle Right-Angled Triangles with Exact Values (Year 11)apply exact values to find a missing triangle side, giving surd answers exactly Right-Angled Triangles with Exact Values (Year 10)the same skill, freshly generated, for review or a Year 10 class Grade 9 Pythagoras' Theorem & Trigonometry worksheetcontrast with calculator-based (decimal) trigonometry

Reach every student

Differentiation

Support

Build a small reference triangle (legs 1, 1, hypotenuse $\sqrt{2}$ ) and another (sides 1, $\sqrt{3}$ , 2) as physical memory aids, rather than memorising the table as abstract symbols.
Practise reciting the table in order (0, 30, 45, 60, 90) repeatedly before attempting mixed-order recall questions.
When applying a ratio to find a side, write out 'opposite = hypotenuse x sin theta' (or the relevant ratio) in full before substituting numbers.

Extension

Derive the 30-60-90 triangle's exact side ratios from scratch, starting from an equilateral triangle of side 2 split by an altitude.
Investigate the exact value of sin(theta) + cos(theta) at theta = 45°, and connect the result to $\sqrt{2}$ .
Preview how these same exact values reappear in the exact values of sin and cos for angles like 120° and 150°, beyond this unit's 0-90° Foundation-tier scope.

Check it stuck

Assessment: exit ticket

A three-question exit ticket sampling recall, reverse lookup, and application to a triangle side.

1. State the exact value of sin 90°.
Answer: 1
2. For 0° ≤ theta ≤ 90°, find theta for which tan theta = 1.
Answer: theta = 45°
3. A right-angled triangle has hypotenuse 10 cm and an angle of 30°. Find the side adjacent to the 30° angle, exactly.
Answer: 5sqrt(3) cm, because adjacent = 10 x cos 30° = 10 x $\sqrt{3}$ /2 = 5sqrt(3).

For the teacher

Teacher notes and timings

Rough timing: Lesson 1 the table and where it comes from (section 1), Lesson 2 applying it to triangle sides plus the exit ticket (section 2).
Curriculum note: DfE GCSE Geometry item 21 is entirely underlined (Foundation, assessed for every student); the closely related sine rule/cosine rule items (22-23) are entirely bold (Higher only) and are a different, harder skill this unit does not touch, and 3D right-angled-triangle applications (the bold clause of item 20) are also excluded.
No figure accompanies this unit: components/StandardFigures.tsx's RightTriangle always computes and displays its hypotenuse as a rounded DECIMAL (via Math.sqrt), which would visually contradict this unit's entire point that the true value is an exact surd, not a decimal, so it is deliberately not used here. Every answer is still fully computed; sketch the two reference triangles (45-45-90 and 30-60-90) on the board to accompany the worked examples.

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