Trig identities: sin²θ + cos²θ = 1
The trig identities are sin²θ + cos²θ ≡ 1, tan θ ≡ sin θ / cos θ, 1 + tan²θ ≡ sec²θ.
What each letter means
- \(\theta\) any angle
- \(\sin^2\theta\) means (sin θ)², the square of sin θ
When to use it
To find cos θ or tan θ from sin θ (or the other way round), to simplify expressions, and to turn a trig equation into one with a single trig function.
Worked example
Solve \(2\cos^2x+3\sin x=3\) for \(0^\circ\le x\le360^\circ\).
- \(2(1-\sin^2x)+3\sin x-3=0\), so \(2\sin^2x-3\sin x+1=0\)
- \((2\sin x-1)(\sin x-1)=0\), so \(\sin x=\tfrac12\) or \(\sin x=1\)
Answer: \(x=30^\circ,\ 90^\circ,\ 150^\circ\)
Common mistake
Forgetting the sign. sin²θ + cos²θ = 1 gives cos θ = ±√(1 − sin²θ): choose the sign from the quadrant.
On your course
| Course | In the exam |
|---|---|
| Edexcel A Level (9MA0) | Not in the Edexcel booklet: learn it |
| Edexcel International A Level | Not in the booklet: learn it |
| AQA or OCR A Level | Check your board’s formula booklet: AQA formulae booklet (PDF) · OCR A Level Maths assessment page |
From our own A Level Maths formula sheets, in our words. Official: Pearson's formulae booklet (PDF) · Pearson's IAL formulae booklet (PDF).
Practise and revise
- Practise A Level Maths practice
- Revise Trigonometry
- Revise IAL P3 trigonometric functions
- Print Edexcel A Level (9MA0) one-page formula sheet
Questions
What is the trig identities formula?
The trig identities are sin²θ + cos²θ ≡ 1, tan θ ≡ sin θ / cos θ, 1 + tan²θ ≡ sec²θ. θ: any angle; sin²θ: means (sin θ)², the square of sin θ.
Is the trig identities given in the exam?
Edexcel A Level (9MA0): not in the Edexcel booklet: learn it. Edexcel International A Level: not in the booklet: learn it. AQA or OCR A Level: check your board’s formula booklet. This comes from our own A Level Maths formula sheets; your teacher has the official booklet.
Where does sin²θ + cos²θ = 1 come from?
A point at angle θ on the unit circle is (cos θ, sin θ). Pythagoras on the radius, which has length 1, gives cos²θ + sin²θ = 1.
Our own wording, examples and card, checked by A Level Math Revision. Not produced or endorsed by Pearson Edexcel, AQA or OCR.