Related angles, arcs and sectors
Mirror a point on the unit circle to see how the sine and cosine of θ and 180° − θ are linked, use the reference angle to find cos 120°, then work out the length of an arc and the area of a sector with the angle in radians.
- UK: Year 12
- France: grade 11
- Germany: grade 10
- Iran: grade 11
- Afghanistan: grade 10
- Syria: grade 10
- Palestine: grade 10
- Ukraine: grade 10
- Russia: grade 10
Guide
For Afghan Grade 10 (chapter 4: relations between the ratios of different angles; chapter 5: arc length and sector area), Iran Grade 11 and A-level. The coloured dots can be dragged; the readouts update live.
Point P is at angle θ on the unit circle. Its mirror image Q in the y-axis is at angle 180° − θ. They are at the same height, so sin (180° − θ) = sinθ, but their x-coordinates are opposite, so cos (180° − θ) = −cosθ. Reflecting in the x-axis, or turning through 180°, gives the other related angles in the same way.
An arc of a circle of radius r that subtends an angle θ in radians has length l = rθ. The sector it cuts off has area A = r2θ2.
The angle must be in radians: 60° = π3. Putting degrees into these formulas gives a wrong answer.