Exact Trigonometric Values (AA SL)
Five angles - \(0,\ \tfrac{\pi}{6},\ \tfrac{\pi}{4},\ \tfrac{\pi}{3},\ \tfrac{\pi}{2}\) - and their multiples give exact sine, cosine and tangent values without a calculator, and IB examiners lean on them constantly in non-calculator papers. This page sets out the values themselves, how to get tangent from them for free, and the sign slip-ups that catch people out once the angle leaves the first quadrant. It's part of the broader Identities & Exact Values topic.
12 questions on this sub-topic.
Sine, cosine and tangent at standard angles
Covered under IB syllabus reference SL3.5: the definitions of \(\cos\theta\) and \(\sin\theta\) from the unit circle, \(\tan\theta=\tfrac{\sin\theta}{\cos\theta}\), and the exact values at \(0,\ \tfrac{\pi}{6},\ \tfrac{\pi}{4},\ \tfrac{\pi}{3},\ \tfrac{\pi}{2}\) and their multiples.
Sine and cosine at standard angles
\[\sin 0=0,\quad \sin\tfrac{\pi}{6}=\tfrac12,\quad \sin\tfrac{\pi}{4}=\tfrac{\sqrt2}{2},\quad \sin\tfrac{\pi}{3}=\tfrac{\sqrt3}{2},\quad \sin\tfrac{\pi}{2}=1\]
\[\cos 0=1,\quad \cos\tfrac{\pi}{6}=\tfrac{\sqrt3}{2},\quad \cos\tfrac{\pi}{4}=\tfrac{\sqrt2}{2},\quad \cos\tfrac{\pi}{3}=\tfrac12,\quad \cos\tfrac{\pi}{2}=0\]
The cosine row is the sine row in reverse - learn one and you already know the other.
Tangent from the ratio
\[\tan\theta=\dfrac{\sin\theta}{\cos\theta}\]
Don't memorise a third row - divide the values you already have, e.g. \(\tan\tfrac{\pi}{3}=\dfrac{\sqrt3/2}{1/2}=\sqrt3\). At \(\tfrac{\pi}{2}\), \(\cos\tfrac{\pi}{2}=0\), so \(\tan\tfrac{\pi}{2}\) is undefined.
For angles beyond the first quadrant, and the full syllabus wording, see Identities & Exact Values.
Worked examples
\(\cos\tfrac{\pi}{4}.\)
(a) Find its exact value.
(b) Find the exact value of \(\tan\tfrac{\pi}{3}\).
Worked solution
(a) \(\tfrac{\pi}{4}\) is a standard angle (\(45^\circ\)), so \(\cos\tfrac{\pi}{4}=\tfrac{\sqrt2}{2}.\) A1
(b) \(\tfrac{\pi}{3}\) is a standard angle (\(60^\circ\)), so \(\tan\tfrac{\pi}{3}=\sqrt3.\) A1
Without a calculator:
(a) Find the exact value of \(\sin\tfrac{\pi}{3}\).
(b) Find the exact value of \(\cos\tfrac{5\pi}{6}\).
Worked solution
(a) \(\tfrac{\pi}{3}\) (\(60^\circ\)) is a standard angle: \(\sin\tfrac{\pi}{3}=\tfrac{\sqrt3}{2}.\) A1
(b) \(\tfrac{5\pi}{6}\) lies in the second quadrant, with reference angle \(\tfrac{\pi}{6}\), where cosine is negative. M1
So \(\cos\tfrac{5\pi}{6}=-\cos\tfrac{\pi}{6}=-\tfrac{\sqrt3}{2}.\) A1
\(\tan\tfrac{4\pi}{3}.\)
(a) Find its exact value.
(b) Find the exact value of \(\cos\tfrac{7\pi}{6}\).
Worked solution
(a) \(\tfrac{4\pi}{3}\): Q3, reference \(\tfrac{\pi}{3}\), tan positive in Q3: M1
\(\tan\tfrac{4\pi}{3}=\sqrt3.\) A1
(b) \(\tfrac{7\pi}{6}\): Q3, reference \(\tfrac{\pi}{6}\), cos negative: \(\cos\tfrac{7\pi}{6}\) M1
\(=-\tfrac{\sqrt3}{2}.\) A1
\(\sin\tfrac{2\pi}{3}.\)
(a) Find its exact value.
(b) Find the exact value of \(\tan\tfrac{5\pi}{6}\).
Worked solution
(a) \(\tfrac{2\pi}{3}\): Q2, reference \(\tfrac{\pi}{3}\), sine positive: M1
\(\sin\tfrac{2\pi}{3}=\tfrac{\sqrt3}{2}.\) A1
(b) \(\tfrac{5\pi}{6}\): Q2, reference \(\tfrac{\pi}{6}\), tan negative: \(\tan\tfrac{5\pi}{6}=-\tfrac{1}{\sqrt3}\) M1
\(=-\tfrac{\sqrt3}{3}.\) A1
Common mistakes
- Swapping the sine and cosine values at \(\tfrac{\pi}{6}\) and \(\tfrac{\pi}{3}\). \(\sin\tfrac{\pi}{6}=\tfrac12\) and \(\cos\tfrac{\pi}{6}=\tfrac{\sqrt3}{2}\) - it's easy to write these the wrong way round under pressure, especially since \(\tfrac{\pi}{3}\) has the values reversed.
- Forgetting the sign changes outside the first quadrant. \(\cos\tfrac{2\pi}{3}\) uses the same reference angle as \(\cos\tfrac{\pi}{3}\), but cosine is negative in the second quadrant - the exact value is \(-\tfrac12\), not \(\tfrac12\).
- Writing \(\tan\tfrac{\pi}{2}\) as a number. Since \(\cos\tfrac{\pi}{2}=0\), the ratio \(\tfrac{\sin\theta}{\cos\theta}\) is undefined there, not zero or a large value.
Ready to practise properly?
12 exact-values questions, marked instantly like the real exam.
Quick answers
Which angles do I need exact trigonometric values for?
\(0,\ \tfrac{\pi}{6},\ \tfrac{\pi}{4},\ \tfrac{\pi}{3}\) and \(\tfrac{\pi}{2}\) (and their multiples around the unit circle) - these five standard angles cover almost every non-calculator trig question at SL.
Do I need to memorise a separate table of exact tangent values?
No - \(\tan\theta=\tfrac{\sin\theta}{\cos\theta}\), so dividing the sine and cosine values you already know is quicker and less error-prone than memorising a third row.