Course content
- Use the identity \(sin^2\ x + cos^2\ x = 1\)
- Use the identity \(tan\ x = \large \frac{sin\ x}{cos\ x} \normalsize \)
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.
Textbook page references
- TeeJay Maths Book N5 pp.201-202
- Zeta National 5+ Maths pp.217-218
- Leckie National 5 Maths pp.287-288
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.
Recommended revision courseA great resource to prepare for your N5 Maths exams. Get £10 discount with code 'Maths.scot'. Learn more |
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.
Recommended student books
Zeta Maths: National 5+ practice bookLeckie: National 5 Maths textbook
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 \(\large\frac{sin\ x^\circ+\ 2\ cos\ x^\circ}{cos\ x^\circ}\)
Recommended revision guides
How to Pass National 5 MathsBrightRED N5 Maths Study Guide
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)
Solve the equation \( 3\tiny\ \normalsize sin\ x=2\tiny\ \normalsize cos\ x \), for \(0 \leq x \lt 360^\circ.\)
Example 8 (non-calculator)
Prove that, for all values of \(x^\circ,\)
\( cos^2\, x\ \left(1+tan^2\,x\right) = 1\)
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 |
Other great resources
Video - Mr Graham Maths |
Video - Mearns Academy |
Notes and videos - Mistercorzi |
Notes - BBC Bitesize |
Notes and exercises - Maths 777 1. Simpler trig identities 2. Harder trig identities |
Practice questions - Maths Hunter |
Exercises - Larkhall Academy Pages 30-31 Ex 7 |
⇦ Nat 5 topic list ⇧ Top of this page
|