National 5 Maths
Trigonometric Identities
Page sections 
- Topic content
- Textbook page numbers
- Trig identities explained
- Worked examples
- Past paper questions
- Worksheets
- Notes and videos
Topic content
- Use the identity \(sin^2\ x + cos^2\ x = 1\)
- Use the identity \(tan\ x = \large \frac{sin\ x}{cos\ x} \normalsize \)
Textbook page numbers
- Zeta National 5+ Maths pp.217-218
- TeeJay Maths Book N5 pp.201-202
- Leckie National 5 Maths pp.287-288
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Zeta National 5+ Maths
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Progressive exercises.
Includes answers.
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Key ideas
-
An identity is not the same as an equation.
- Equations can be solved to find the value, or values, of the variable that make it true.
- Identities are always true, for every value of the variable. They are statements of fact.
- The two Nat 5 trig identities are not on the formulae list. You will need to learn them.
Proof: \(sin^2 x + cos^2 x = 1\)
You don't need to learn this proof, but some of you will find it interesting to know why the identity is true.
Imagine a right-angled triangle with \(x^\circ\) as one of its angles. We know that \(sin\ x=\large\frac{opp}{hyp}\normalsize\) and \(cos\ x=\large\frac{adj}{hyp}\normalsize.\)
Now let's use this to prove that the identity is true for any acute angle \(x^\circ.\)
$$ \begin{eqnarray} sin^2 x + cos^2 x &=& \left(\frac{opp}{hyp}\right)^2+\left(\frac{adj}{hyp}\right)^2 \\[9pt] &=& \frac{opp^2+adj^2}{hyp^2} \\[9pt] &=& \frac{hyp^2}{hyp^2} \:\:\small\textsf{(by Pythagoras)}\normalsize \\[9pt] &=& 1 \end{eqnarray} $$
In fact, this identity is also true for any non-acute angle, but that proof is beyond National 5 level.
Proof: \(tan\ x = \large \frac{sin\ x}{cos\ x} \normalsize \)
Again, you don't have to learn this proof, but the techniques that it uses are useful.
We will start with the right hand side and simplify it to the left hand side.
$$ \begin{eqnarray} \frac{sin\ x}{cos\ x} &=& \frac{opp}{hyp} \div \frac{adj}{hyp} \\[9pt] &=& \frac{opp}{hyp} \times \frac{hyp}{adj}\\[9pt] &=& \frac{opp}{\cancel{hyp}} \times \frac{\cancel{hyp}}{adj} \\[9pt] &=& \frac{opp}{adj} \\[9pt] &=& tan\ x \end{eqnarray} $$
This identity is also true for any non-acute angle, but again that proof is beyond N5.
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Example 1 (non-calculator)
SQA National 5 2016 Paper 1 Question 11
Simplify: \( tan^2 x^\circ\ cos^2\ x^\circ \)
Show your working.
Example 2 (non-calculator)
SQA National 5 2018 Paper 1 Question 18
Express \( sin x^\circ\ cos x^\circ\ tan x^\circ \) in its simplest form. Show your working.
Example 3 (calculator)
SQA National 5 2019 Paper 2 Question 17
Expand and simplify \( \left(sin x^\circ+cos x^\circ\right)^2 \)
Show your working.
Example 4 (calculator)
SQA National 5 2021 Paper 2 Question 16
Expand and simplify \(cos\ x^\circ\ \left(tan\ x^\circ +1\right)\)
Show your working.
Example 5 (calculator)
SQA National 5 2022 Paper 2 Question 13
Simplify \(\large\frac{sin\ x^\circ+\ 2\ cos\ x^\circ}{cos\ x^\circ}\)
Example 6 (calculator)
SQA National 5 2023 Paper 2 Question 13
Simplify \(2\,sin^2\,x^\circ +2\,cos^2\,x^\circ\small.\) Show your working.
Example 7 (calculator)
SQA National 5 2024 Paper 2 Question 16
Express \(3\,cos^2\,x^\circ -1\) in the form \(a+b\,sin^2\,x^{\circ}\small.\) Show your working.
Example 8 (calculator)
Solve the equation \( 3\tiny\ \normalsize sin\ x=2\tiny\ \normalsize cos\ x \), for \(0 \leq x \lt 360^\circ.\)
Example 9 (non-calculator)
Prove that, for all values of \(x^\circ,\)
\( cos^2\, x\ \left(1+tan^2\,x\right) = 1\)
Buy N5 Maths practice papers
Zeta: Five Practice Papers TOP CHOICECGP: N5 Maths Exam Practice
Leckie: Revision and Practice
Hodder: N5 Maths Practice Papers
Past paper questions
|
• 2016 Paper 1 Q11 • 2018 Paper 1 Q18 • 2019 Paper 2 Q17 (with expansion) • 2021 Paper 2 Q16 • 2022 Paper 2 Q13 • 2023 Paper 2 Q13 • 2024 Paper 2 Q16 |
| 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 identities worksheets
| Maths Hunter worksheet • Practice questions (no answers) |
|
Corbettmaths worksheet • Trig identities (Answers) |
|
National5Maths.co.uk worksheet • Trig identities (with solutions) |
|
CJ Maths worksheet • Trig identities (no answers) |
| Larkhall Academy exercises • Pages 30-31 Ex 7 (no answers) |
Buy N5 Maths revision guides
How to Pass N5 Maths TOP CHOICEBrightRED: N5 Maths Study Guide
CGP: N5 Maths Revision Guide
Notes and videos
| Video - Mr Graham Maths |
| Video - Mearns Academy |
| Notes and videos - Mistercorzi |
| Notes - BBC Bitesize |
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