Trig identities: sin²θ + cos²θ = 1
The trig identities are sin²θ + cos²θ = 1 and tan θ = sin θ / cos θ.
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.
AA also uses the double angle identities sin 2θ = 2 sin θ cos θ and cos 2θ = cos²θ − sin²θ; AA HL adds 1 + tan²θ = sec²θ.
Worked example
\(\theta\) is obtuse and \(\sin\theta=\tfrac35\). Find \(\cos\theta\) and \(\tan\theta\).
- \(\cos^2\theta=1-\tfrac{9}{25}=\tfrac{16}{25}\)
- \(\theta\) is obtuse, so \(\cos\theta<0\)
Answer: \(\cos\theta=-\tfrac45\), \(\tan\theta=-\tfrac34\)
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 |
|---|---|
| AA SL | In the IB formula booklet |
| AA HL | In the IB formula booklet |
| AI HL | In the IB formula booklet |
From our own IB Maths formula sheets, in our words. The IB gives its booklet to schools, so we checked against our own copy: your teacher has the official one. Official: IB DP Mathematics page.
Practise and revise
- Practise AA SL trigonometric identities questions
- Practise AA SL trigonometric equations questions
- Revise Trig identities (AA SL)
- Revise Compound angles (AA HL)
- Tool Graph of y = sin x
- Print AA SL one-page formula sheet
Questions
What is the trig identities formula?
The trig identities are sin²θ + cos²θ = 1 and tan θ = sin θ / cos θ. θ: any angle; sin²θ: means (sin θ)², the square of sin θ.
Is the trig identities given in the exam?
AA SL: in the IB formula booklet. AA HL: in the IB formula booklet. AI HL: in the IB formula booklet. This comes from our own IB 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 IB Math Revision. Not produced or endorsed by the International Baccalaureate Organization.