Identities & Exact Values (AA SL)
Trigonometric identities let you rewrite one trig expression in terms of another without ever finding the angle itself, and exact values let you state results like \(\sin\tfrac{\pi}{3}\) as a clean surd instead of a rounded decimal. This topic covers the Pythagorean identity, the double angle identities for sine and cosine, and the exact values of sine, cosine and tangent at the special angles \(0,\ \tfrac{\pi}{6},\ \tfrac{\pi}{4},\ \tfrac{\pi}{3},\ \tfrac{\pi}{2}\).
What the syllabus says
This topic maps onto two points in the official IB Analysis & Approaches syllabus.
| Code | Syllabus content |
|---|---|
| SL3.5 | Definition of \(\cos\theta\), \(\sin\theta\) in terms of the unit circle, including relationships between angles in different quadrants. Definition of \(\tan\theta\) as \(\dfrac{\sin\theta}{\cos\theta}\). Exact values of trigonometric ratios of \(0,\ \tfrac{\pi}{6},\ \tfrac{\pi}{4},\ \tfrac{\pi}{3},\ \tfrac{\pi}{2}\) and their multiples. Extension of the sine rule to the ambiguous case. |
| SL3.6 | The Pythagorean identity \(\cos^2\theta+\sin^2\theta=1\). Double angle identities for sine and cosine. The relationship between trigonometric ratios - for example, given \(\sin\theta\), finding possible values of \(\tan\theta\) without finding \(\theta\). |
Both are core AA SL syllabus points, examined mainly on Paper 1.
Key terms
Five words worth knowing cold before you touch the formulas below - each with a worked example showing exactly what it means.
What is the Pythagorean identity?
The Pythagorean identity states that \(\sin^2\theta+\cos^2\theta=1\) for every angle \(\theta\). It comes directly from Pythagoras' theorem applied to the unit circle, and lets you find one ratio from the other.
e.g. If \(\sin\theta=\tfrac35\), then \(\cos^2\theta=1-\tfrac{9}{25}=\tfrac{16}{25}\), so \(\cos\theta=\tfrac45\) (for \(\theta\) acute).
What are "exact values"?
Exact values are the surd or fraction forms of \(\sin\), \(\cos\) and \(\tan\) at the special angles \(0,\ \tfrac{\pi}{6},\ \tfrac{\pi}{4},\ \tfrac{\pi}{3},\ \tfrac{\pi}{2}\) - values you're expected to know or derive without a calculator, rather than round to a decimal.
e.g. \(\tan\tfrac{\pi}{3}=\dfrac{\sin(\pi/3)}{\cos(\pi/3)}=\dfrac{\sqrt3/2}{1/2}=\sqrt3.\)
What is a double angle identity?
A double angle identity rewrites \(\sin2\theta\) or \(\cos2\theta\) in terms of \(\sin\theta\) and \(\cos\theta\) - useful when a question gives you information about \(\theta\) but asks about \(2\theta\).
e.g. If \(\sin\theta=\tfrac35\) and \(\cos\theta=\tfrac45\), then \(\sin2\theta=2(\tfrac35)(\tfrac45)=\tfrac{24}{25}.\)
What is the unit circle?
The unit circle is a circle of radius 1 centred at the origin. For an angle \(\theta\) measured from the positive \(x\)-axis, the point on the circle has coordinates \((\cos\theta,\sin\theta)\) - this is the formal definition behind every trig identity.
e.g. At \(\theta=\tfrac{\pi}{2}\), the point on the unit circle is \((0,1)\), so \(\cos\tfrac{\pi}{2}=0\) and \(\sin\tfrac{\pi}{2}=1\).
What does "relationship between trig ratios" mean?
It means finding one trig ratio from another - like \(\tan\theta\) from \(\cos\theta\) - using the Pythagorean identity and the definition \(\tan\theta=\sin\theta/\cos\theta\), all without ever calculating \(\theta\) itself.
e.g. If \(\cos\theta=\tfrac34\) and \(\theta\) is acute, \(\sin\theta=\sqrt{1-\tfrac{9}{16}}=\dfrac{\sqrt7}{4}\), so \(\tan\theta=\dfrac{\sqrt7/4}{3/4}=\dfrac{\sqrt7}{3}.\)
Key formulas
A handful of identities, plus one table of special-angle values, cover almost every question on this topic. The tables below summarise them at a glance - the explanations underneath go into more depth on each one.
Formula reference
The Pythagorean identity and both double angle identities are printed in the formula booklet; the exact-value table is prior knowledge you're expected to know or be able to derive.
| Formula | Used for | Booklet? |
|---|---|---|
| \(\sin^2\theta+\cos^2\theta=1\) | Pythagorean identity | ✓ Yes |
| \(\sin2\theta=2\sin\theta\cos\theta\) | Double angle identity for sine | ✓ Yes |
| \(\cos2\theta=\cos^2\theta-\sin^2\theta=2\cos^2\theta-1=1-2\sin^2\theta\) | Double angle identity for cosine (three equivalent forms) | ✓ Yes |
| \(\tan\theta=\dfrac{\sin\theta}{\cos\theta}\) | Definition of tangent | Not in booklet - prior knowledge |
Exact values of special angles
These values come up constantly on Paper 1 and are worth memorising rather than re-deriving each time.
| \(\theta\) | \(\sin\theta\) | \(\cos\theta\) | \(\tan\theta\) |
|---|---|---|---|
| \(0\) | \(0\) | \(1\) | \(0\) |
| \(\tfrac{\pi}{6}\) | \(\tfrac12\) | \(\tfrac{\sqrt3}{2}\) | \(\tfrac{1}{\sqrt3}\) |
| \(\tfrac{\pi}{4}\) | \(\tfrac{\sqrt2}{2}\) | \(\tfrac{\sqrt2}{2}\) | \(1\) |
| \(\tfrac{\pi}{3}\) | \(\tfrac{\sqrt3}{2}\) | \(\tfrac12\) | \(\sqrt3\) |
| \(\tfrac{\pi}{2}\) | \(1\) | \(0\) | undefined |
The Pythagorean identity
This identity is the single most useful tool for switching between \(\sin\theta\) and \(\cos\theta\) without finding \(\theta\).
The core identity
\[\sin^2\theta+\cos^2\theta=1\]
True for every angle \(\theta\) - it's the Pythagorean theorem applied to the unit circle.
Finding one ratio from the other
\[\sin\theta=\pm\sqrt{1-\cos^2\theta}\]
Decide the sign from the quadrant \(\theta\) is in, or from information like "\(\theta\) is acute".
Double angle identities
These rewrite \(\sin2\theta\) and \(\cos2\theta\) in terms of the single-angle ratios.
Double angle for sine
\[\sin2\theta=2\sin\theta\cos\theta\]
Only one form - always the product of sine and cosine, doubled.
Double angle for cosine
\[\cos2\theta=2\cos^2\theta-1=1-2\sin^2\theta\]
Pick whichever form matches the information you're given - both are equivalent via the Pythagorean identity.
Worked examples
Two full exam-style questions, marked exactly like the real thing. Try each one yourself before checking the worked solution.
\(\theta=\dfrac{\pi}{6}.\)
(a)(i) Write down the exact value of \(\sin\theta\).
(a)(ii) Write down the exact value of \(\cos\theta\).
(b) Given \(\cos\theta=\tfrac35\) with \(\theta\) acute, find \(\sin\theta\).
Worked solution
(a)(i) \(\sin\tfrac{\pi}{6}=\tfrac12.\) A1
(a)(ii) \(\cos\tfrac{\pi}{6}=\tfrac{\sqrt3}{2}.\) A1
(b) Apply the Pythagorean identity \(\sin^2\theta+\cos^2\theta=1\): \(\sin^2\theta=1-\tfrac{9}{25}=\tfrac{16}{25}.\) M1
Since \(\theta\) is acute, take the positive root: \(\sin\theta=\tfrac45.\) A1
Solve \(2\cos^2\theta-1=0\) for \(0^\circ\le\theta\le360^\circ\).
(a)(i) State the smallest solution.
(a)(ii) State the second solution.
(a)(iii) State the third solution.
(a)(iv) State the largest solution.
Worked solution
\(\cos^2\theta=\tfrac12\Rightarrow\cos\theta=\pm\dfrac{\sqrt2}{2}.\) M1 \(\theta=45^\circ,135^\circ,225^\circ,315^\circ.\) A1 A1 A1 A1
Common mistakes
The four slip-ups that account for most of the marks lost on this topic - worth reading before you start practising, not just after you get one wrong.
- Forgetting the \(\pm\) when taking a square root. \(\sin\theta=\pm\sqrt{1-\cos^2\theta}\) - dropping the negative option loses solutions unless the question restricts \(\theta\) (e.g. "acute").
- Writing \(\cos2\theta=2\cos\theta\). Doubling the angle is not the same as doubling the ratio - the correct identity is \(\cos2\theta=2\cos^2\theta-1\) (or an equivalent form).
- Mixing up \(\sin\tfrac{\pi}{3}\) and \(\cos\tfrac{\pi}{3}\). It's easy to swap \(\tfrac12\) and \(\tfrac{\sqrt3}{2}\) between sine and cosine - sketching the unit circle helps keep them straight.
- Not checking the quadrant before assigning a sign. The sign of \(\sin\theta\), \(\cos\theta\) or \(\tan\theta\) depends on which quadrant \(\theta\) is in - always check this before writing a final answer.
Using your GDC
Every step below is a real button sequence, not a vague "use your calculator" hint - covering the TI-84 Plus, TI-Nspire, and Casio fx-9860/fx-CG50. Pick your model to filter down to just the steps that apply to you.
Get exact fractions and surds instead of long decimals, or convert a decimal answer back to a tidy fraction to check it.
- After a calculation press MATH → 1:►Frac then ENTER to turn the last answer into a fraction.TI-84
- Press ctrl → ENTER (instead of ENTER) for the exact form; or menu → Number → Approximate to Fraction.Nspire
- Press the F↔D key to toggle the answer between fraction and decimal; Math input mode (SET UP) shows surds and stacked fractions.Casio
- Use this to confirm a messy decimal is really a nice value like \(\tfrac78\) or \(\sqrt2\).
Tip: Convert to a decimal only at the very end - keep exact values through the working.
See the full GDC guide for more calculator models and topics.
Ready to practise properly?
Identities and exact value questions, marked instantly like the real exam.
Quick answers
The questions students on this topic ask most often.
What is the Pythagorean identity and when do I use it?
The Pythagorean identity is sin squared theta plus cos squared theta equals 1. Use it whenever you know one of sin theta or cos theta and need the other, without ever finding theta itself.
How do I find sin theta from cos theta without finding theta?
Rearrange the Pythagorean identity to sin squared theta equals 1 minus cos squared theta, then take the square root. You'll need to decide the sign (positive or negative) from the quadrant the angle is in, or from information given in the question such as theta being acute.
What are the double angle identities?
Sin 2 theta equals 2 sin theta cos theta, and cos 2 theta has three equivalent forms: cos squared theta minus sin squared theta, 2 cos squared theta minus 1, or 1 minus 2 sin squared theta. Which form to use depends on what information the question gives you.
Can I use my GDC for this topic?
Yes, for checking work or evaluating a decimal, but Paper 1 questions on exact values and identities are set without a calculator - you're expected to know the special-angle values and apply the identities algebraically. See the GDC guide for how to display exact fractions and surds.
Sub-topics
Identities & Exact Values broken down into its individual skills, each with its own focused page.
Related topics
More Geometry & Trigonometry topics from the same AA SL syllabus unit, in case you want to keep going.