The unit circle — exact values and identities (IB Math AA SL)

Key ideas

• sin of 0, π/6, π/4, π/3, π/2 is 0, 1/2, √2/2, √3/2, 1. cos is the same list in reverse: 1, √3/2, √2/2, 1/2, 0.
• tan of 0, π/6, π/4, π/3 is 0, √3/3 (= 1/√3), 1, √3, and tan π/2 is undefined.
• Signs: in the first quadrant all three are positive, in the second only sin, in the third only tan, in the fourth only cos.
• Symmetry: sin(π − θ) = sin θ, cos(π − θ) = −cos θ, sin(−θ) = −sin θ and cos(−θ) = cos θ.
• Every point (cos θ, sin θ) lies on x² + y² = 1, which is why sin²θ + cos²θ = 1.

Worked example

Solve 2 sin x = −√3 for 0 ≤ x ≤ 2π.

1. sin x = −√3/2.
2. The reference angle is π/3, because sin π/3 = √3/2.
3. sin is negative in the third and fourth quadrants.
4. Third quadrant: x = π + π/3 = 4π/3. Fourth quadrant: x = 2π − π/3 = 5π/3.

Answer: x = 4π/3 or x = 5π/3.

Exam-style SL AA questions on this topic

• The Unit Circle and Exact Values of Trigonometric Ratios questions — 27 questions with mark schemes, plus free worked examples
• All SL AA Geometry and Trigonometry questions by topic

See it on a graph

• Graph of y = sin x
• Graph of y = cos x
• Open the free Grapher

Maths IA ideas that use the unit circle

• How far off do you end up? Navigating a route by bearings
• The hidden angles of a hip roof: 3D trigonometry
• Which method measures a building's height most accurately?
• Measuring the area of an irregular field with triangles
• All SL AA IA ideas

Related in Geometry and Trigonometry

• Trigonometric identities — Pythagorean and double-angle
• Solving trigonometric equations in a given interval
• Sine rule, cosine rule, and area of a triangle

Revise The unit circle for the exam

• SL AA revision notes — every unit summarised, with the key results to learn
• SL AA past papers — specimen papers and timed practice papers
• Formulas for Geometry and Trigonometry — on the one-page SL AA formula sheet (free PDF), with the ones not in the IB booklet marked
• SL AA formula booklet by topic — every formula in the IB booklet's notation, with a free PDF
• IB Maths prediction papers — full papers in the IB format, with markschemes

TOK link: How far should we trust tools whose workings we cannot check?

Stretch yourself: original geometry problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S18 · Problem S74 · Problem S17 · Problem S73. All 22 →

Formulas in this topic: Arc length · Area of a sector. All formulas A–Z