Angles of any size, the unit circle and radians
Turn a point round the unit circle by angles bigger than 90°, negative ones included, read sine and cosine from its position, measure the angle in radians and find exact values such as sin7π6.
📖 Start here: read the steps of the “Lesson” in order (1, 2, 3 …) and answer each question. Then continue with “Practice”.
- UK: Year 12
- France: grade 11
- Germany: grade 10
- Iran: grade 10
- Afghanistan: grade 10
- Syria: grade 10
- Palestine: grade 10
- Ukraine: grade 10
- Russia: grade 10
Guide
For Afghan Grade 10 (chapter 4, trigonometric functions), Iran Grade 10–11 and A-level. Angles are measured anticlockwise from the positive x-axis; a negative angle turns clockwise. coterminal angles end on the same ray and differ by a multiple of 360°.
On the unit circle (radius 1) the point at angle θ has coordinates (cosθ,sinθ): cos is the x-coordinate and sin the height, and both can be negative. In the pictures a circle of radius 3 squares stands for radius 1.
A radian is the angle that cuts off an arc as long as the radius. A half turn is π radians: 180° = π, 90° = π2, 30° = π6.
Exact values come from the reference angle, the acute angle between the ray and the x-axis. The size of sin and cos is the same as for the reference angle; the quadrant gives the sign.