Unit Circle
Unit circle exact values are the precise coordinates and trigonometric ratios for standard angles on a circle of radius 1 centred at the origin. These standard angles—0°, 30°, 45°, 60°, and 90° (and their radian equivalents)—produce clean fractional or radical expressions rather than decimal approximations. The unit circle method extends these values to all 4 quadrants using reference angles and the ASTC rule (All, Sine, Tangent, Cosine positive).
Why it matters
Exact trigonometric values appear throughout engineering, physics, and advanced mathematics where precision matters more than decimal approximations. In signal processing, engineers use sin(π/4) = √22 for 45° phase shifts in electronic circuits. Architecture relies on cos(60°) = 12 for calculating roof angles and structural loads. Computer graphics engines use these values for rotating objects—sin(90°) = 1 produces perfect 90° rotations without rounding errors. At A-level and university, exact values become essential for calculus integration, where ∫sin(x)dx from 0 to π/3 equals 12 exactly. Physics equations for wave interference and oscillations depend on exact trigonometric ratios. GCSE Higher papers frequently test these values, whilst Further Maths extends them to complex exponential forms.
How to solve unit circle
Unit Circle — Exact Values
- On the unit circle, cos θ = x-coordinate and sin θ = y-coordinate.
- Memorise Q1 values: 30° (½, √32), 45° (√22, √22), 60° (√32, ½).
- Use ASTC to get the sign in other quadrants: All, Sine, Tangent, Cosine are positive.
- Reference angle = acute angle to the x-axis; signs come from the quadrant.
Example: sin(150°) = +sin(30°) = 12 (Q2, sine positive).
Worked examples
Find the exact value of sin(90°).
Answer: 1
- Recall the standard value of sin at 90° → sin(90°) — The angles 0°, 30°, 45°, 60°, and 90° are called *standard angles*. Their sin, cos, and tan values are memorised because they appear over and over in trigonometry.
- Look up sin(90°) → sin(90°) = 1 — You can derive this from a 30-60-90 or 45-45-90 right triangle, or read it off the unit circle diagram.
Find the exact value of tan(30°).
Answer: √33
- Find the reference angle for 30° → reference = 30° — The reference angle is the acute angle between the terminal side and the nearest x-axis. For 30° in Q1, the reference is 30°.
- Evaluate tan(30°) from the standard-angle table → tan(30°) = √33 — The reference angle is always in Q1, so use the memorised values.
- Apply the sign for Q1 using ASTC → tan(30°) = √33 — In Quadrant 1 all three functions (sin, cos, tan) are positive.
Find the exact value of cos(π/2).
Answer: 0
- Convert π/2 radians to degrees → π/2 = 90° — Multiply radians by 180/π to convert to degrees. The standard unit-circle angles have clean degree equivalents.
- Read cos(90°) from the unit circle → cos(π/2) = 0 — This is an axial angle, where the function evaluates to 0, 1, or is undefined.
Common mistakes
- Writing sin(30°) = 0.5 instead of 1/2, missing that exact form is required rather than decimal approximation
- Calculating tan(45°) = 1.414 instead of 1, confusing the tangent value with the sine or cosine at that angle
- Getting cos(150°) = √3/2 instead of -√3/2, forgetting that cosine is negative in the second quadrant
- Converting π/6 radians as 30 degrees instead of 30°, dropping the degree symbol in the final answer