National 5 Maths
Trigonometric Equations

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Page sections

• Topic content
• Textbook page numbers
• Trig equations explained
• Worked examples
• Past paper questions
• Worksheets
• Notes and videos

Topic 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 numbers

• Zeta National 5+ Maths pp.214-216
• TeeJay Maths Book N5 pp.196-203
• Leckie National 5 Maths pp.272-280

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Zeta National 5+ Maths
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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 acute angle \(a^\circ\) at the centre.

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

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

These four angles are what we call related.

Example 1:  \(a\) = 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:  \(a\) = 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:  \(a\) = 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 quadrant 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

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 interval, 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 letter \(a\) for the related acute angle. It is important not to use \(x\) for this, as the acute angle is not always one of the solutions.

Once we know the related acute angle, we will use it to find the values of \(x\small.\)

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Example 1 (calculator)

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

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Example 2 (calculator)

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

Show solution

Example 3 (calculator)

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

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Example 4 (calculator)

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

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Example 5 (calculator)

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

Show solution

Example 6 (calculator)

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

Show solution

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.

Show solution

Example 8 (calculator)

SQA National 5 Maths 2014 P2 Q12

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

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Example 9 (calculator)

SQA National 5 Maths 2016 P2 Q14

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

Show solution

Example 10 (calculator)

SQA National 5 Maths 2018 P2 Q8

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

Show solution

Example 11 (calculator)

SQA National 5 Maths 2019 P2 Q14

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

Show solution

Example 12 (calculator)

SQA National 5 Maths 2022 P2 Q9

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

Show solution

Example 13 (calculator)

SQA National 5 Maths 2024 P2 Q11

Solve the equation \( 17\tiny\ \normalsize sin\ x^\circ+1=9 \), for \(0 \leq x \lt 360.\)

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Buy N5 Maths practice papers

Zeta: Five Practice Papers   TOP CHOICE
CGP: N5 Maths Exam Practice 
Leckie: Revision and Practice 
Hodder: N5 Maths Practice Papers 

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 • 2024 Paper 2 Q11 • 2025 Paper 2 Q14

Standard Grade: Credit (1986–2013) • Exam questions and answers • More exam questions and answers

Intermediate 2 (2000–2015) • Exam questions (with answers)

Buy our favourite N5 textbook

Zeta National 5+ Maths
Clear and comprehensive.
Progressive exercises.
Includes answers.
Buy from Zeta Press 

Trig equations worksheets

Maths.scot worksheet • Trig equations (Answers)

Essential Skills worksheet • Trig equations (Answers)

Madras College worksheet • Trig equations (with answers)

Maths Hunter worksheet • Practice questions (no answers)

Inverclyde Academy worksheet • Trig equations (no answers)

CJ Maths worksheet • Trig equations (no answers)

Larkhall Academy exercises • Pages 29-30 Ex 4-5 (no answers)

Buy N5 Maths revision guides

How to Pass N5 Maths    TOP CHOICE
BrightRED: N5 Maths Study Guide 
CGP: N5 Maths Revision Guide 

Notes and videos

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

Video - Direct Tutoring

PowerPoint - MathsRevision.com

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

Notes - Maths4Scotland

Notes - National5.com

Worked example - D R Turnbull

Examples - Maths Mutt

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