Trig Equations

Course content

Understand how sin, cos and tan are defined for angles from 0° to 360°
Understand period and related angles
Solve basic trigonometric equations.

Textbook page references

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Related angles

The trig functions sin, cos and tan can be defined for angles greater than 90° by thinking about this diagram:

The four quadrants contain congruent right-angled triangles with the angle \(x^\circ\) at the centre.

The actual angle is measured anticlockwise from the 0° line.

1st quadrant: The actual angle is \(x^\circ.\)
2nd quadrant: The actual angle is half a turn and then back \(x^\circ.\) So that's 180–\(x^\circ.\)
3rd quadrant: The actual angle is half a turn and then another \(x^\circ\). So that's 180+\(x^\circ.\)
4th quadrant: The actual angle is a full turn and then back \(x^\circ.\) So that's 360–\(x^\circ.\)

These four angles are what we call related.

Example 1: \(x\) = 30°

Check this yourself on your calculator:

sin 30° = \(1 \over 2\)
sin (180–30)° = sin 150° = \(1 \over 2\)
sin (180+30)° = sin 210° = \(-\frac{1}{2}\)
sin (360–30)° = sin 330° = \(-\frac{1}{2}\)

Apart from the negatives, they're all equal. That's because these four angles are related.

Example 2: \(x\) = 45°

Check this yourself on your calculator:

cos 45° = \(\frac{\sqrt{2}}{2}\)
cos (180–45)° = cos 135° = \(-\frac{\sqrt{2}}{2}\)
cos (180+45)° = cos 225° = \(-\frac{\sqrt{2}}{2}\)
cos (360–45)° = cos 315° = \(\frac{\sqrt{2}}{2}\)

Again, these are related angles, so they're equal apart from the negatives.

Example 3: \(x\) = 60°

Check this yourself on your calculator:

tan 60° = \(\sqrt{3}\)
tan (180–60)° = tan 120° = \(-\sqrt{3}\)
tan (180+60)° = tan 240° = \(\sqrt{3}\)
tan (360–60)° = tan 300° = \(-\sqrt{3}\)

Same again: they're equal apart from the negatives, because they are related angles.

ASTC diagram

This is also called a CAST diagram, but we prefer ASTC as it keeps the letters in the correct quadrant order.

Remember it with one of these:

All Sinners Take Care.
All Students Take Calculus.
Add Sugar To Coffee.
A Smart Trig Class.
All Stations To Central.

Here is a simpler version of the diagram:

If you want to understand why some of the values for related angles are positive and some are negative, that's easy. Just go back to your study of the basic trig graphs and take a look at when they are above or below the \(x\)-axis. Above is positive and below is negative – simple!

Trig equations have an infinity of solutions because the wave-like graphs of these functions repeat endlessly.

For this reason, Nat 5 exams ask for such equations to be solved within a given domain, often \(0 \leq x \lt 360.\)

When solving any trig equation, we recommend always finding the related acute angle. This means the 1st quadrant (0° to 90°) angle that is related to the solutions.

We will use the Greek letter alpha (\(\alpha\)) for the related acute angle.

Once we know the related acute angle, we will use it to find the solutions.

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Example 1 (calculator)

Solve the equation \( cos\ x^\circ=0.5 \), for \(0 \leq x \lt 360.\)

Example 2 (calculator)

Solve the equation \( cos\ x^\circ=-0.5 \), for \(0 \leq x \lt 360.\)

Example 3 (calculator)

Solve the equation \( sin\ x^\circ=-0.4 \), for \(0 \leq x \lt 360.\)

Recommended student books

Zeta Maths: National 5+ practice book

Leckie: National 5 Maths textbook

Example 4 (calculator)

Solve the equation \( tan\ x^\circ=5 \), for \(0 \leq x \lt 360.\)

Example 5 (calculator)

Solve the equation \( 9\tiny\ \normalsize tan\ x^\circ-1=6 \), for \(0 \leq x \lt 360.\)

Example 6 (calculator)

Solve the equation \( 3\tiny\ \normalsize sin\ x^\circ+5=4 \), for \(0 \leq x \lt 360.\)

Recommended revision guides

How to Pass National 5 Maths

BrightRED N5 Maths Study Guide

Example 7 (non-calculator)

Write the following in order of size, starting with the smallest.

\(sin\ 200^\circ\) \(sin\ 160^\circ\) \(sin\ 180^\circ\)

Justify your answer.

Example 8 (calculator)

SQA National 5 Maths 2014 P2 Q12

Solve the equation \( 11\tiny\ \normalsize cos\ x^\circ-2=3 \), for \(0 \leq x \leq 360.\)

Example 9 (calculator)

SQA National 5 Maths 2016 P2 Q14

Solve the equation \( 2\tiny\ \normalsize tan\ x^\circ+5=-4 \), for \(0 \leq x \leq 360.\)

N5 Maths practice papers

Non-calculator papers and solutions

Calculator papers and solutions

Example 10 (calculator)

SQA National 5 Maths 2018 P2 Q8

Solve the equation \( 7\tiny\ \normalsize sin\ x^\circ+2=3 \), for \(0 \leq x \lt 360.\)

Example 11 (calculator)

SQA National 5 Maths 2019 P2 Q14

Solve the equation \( 5\tiny\ \normalsize cos\ x^\circ+2=1 \), for \(0 \leq x \leq 360.\)

Example 12 (calculator)

SQA National 5 Maths 2022 P2 Q9

Solve the equation \( 3\tiny\ \normalsize sin\ x^\circ+4=6 \), for \(0 \leq x \leq 360.\)

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Maths.scot worksheet

• Trig equations worksheet
• Answer sheet
• See all National 5 Maths worksheets

Past paper questions

• All past paper questions by topic

• 2013 Specimen Paper 2 Q10(b)
• 2014 Paper 2 Q12
• 2015 Paper 1 Q9 (ordering)
• 2016 Paper 2 Q14
• 2017 Specimen Paper 2 Q12
• 2017 Paper 2 Q15
• 2018 Paper 1 Q12 (related angles)
• 2018 Paper 2 Q8
• 2019 Paper 2 Q14
• 2021 Paper 2 Q14
• 2022 Paper 2 Q9
• 2023 Paper 1 Q11 (related angles)
• 2023 Paper 2 Q11

Other great resources

Video - Mr Graham Maths

Videos - Larbert Maths
1. Introduction to trig equations
2. Trig equations with negatives
3. Rearranging and solving

Videos - Corbett Maths
1. Introduction to trig equations
2. Solving trig equations
3. Equations requiring trig identities

Notes and videos - Mistercorzi

PowerPoint - MathsRevision.com

• Detailed notes - BBC Bitesize
• Test yourself - BBC Bitesize

Notes - Maths4Scotland

Notes - National5.com

Worked example - D R Turnbull

Lesson notes - Maths 777
1. Basic trig equations
2. Trig equations: various domains
2. Trig equations: rearranging

Examples - Maths Mutt

Practice questions - Maths Hunter

Worksheet - Inverclyde Academy

Worksheet - Brannock HS
Essential Skills 15 (Answers)

Worksheet - St Andrew's Academy
Exercise on page 31

Exercises - Larkhall Academy
Pages 29-30 Ex 4-5

Click here to study the trig equations notes on National5.com.

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