Direction of rotation
An angle in standard position has its vertex at the origin and its initial side along the positive x-axis. Rotating the terminal side anticlockwise gives a positive angle; rotating it clockwise gives a negative angle. Because a full turn is 360°, angles that differ by a whole number of turns share the same terminal side. Such angles are called coterminal. For example, 30°, 390° and −330° are all coterminal.
Degrees and radians
One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. Since a full circle is 2π radians and also 360°, we get the conversion below.
Key formula /
180° = π radians, so degrees → radians: × π/180 and radians → degrees: × 180/π. Coterminal angles: θ ± 360°k (or θ ± 2πk).
To reduce a large angle to the range 0° to 360°, subtract or add multiples of 360°. For a negative angle, add 360° repeatedly until the result is positive.
Radians become the natural unit once we start differentiating trigonometric functions, so it is worth being fluent with the common conversions such as 30° = π/6, 45° = π/4, 60° = π/3 and 90° = π/2. When a problem gives an angle greater than one full turn, its trigonometric ratios are identical to those of its reduced coterminal angle, which is exactly why reducing the angle first saves effort and avoids mistakes.
Worked example
Convert 210° to radians and express −60° as a positive coterminal angle in the range 0° to 360°.
210° = 210 × π/180 = 7π/6 radians. For −60°, add 360°: −60° + 360° = 300°, which lies in the required range and shares the same terminal side.
Remember
- Anticlockwise = positive, clockwise = negative.
- Coterminal angles differ by multiples of 360° (2π).
- Keep π in exact answers; use π ≈ 3.142 only when a decimal is asked.