Radians and the unit circle

Lay a circle's radius along its edge and the angle it covers is one radian, about 57.3°. Just over six of them go round the circle.

Measuring angles with the radius

Degrees are a human choice: 360 parts of a turn, a number inherited from Babylonian astronomy. Radians come from the circle itself. OpenStax Precalculus defines one radian as the central angle whose arc is equal in length to the radius. Because the circumference of a circle is 2πr, exactly 2π radius-lengths fit round it, so a full turn is 2π radians. That fixes the conversion: 2π rad = 360°, or π rad = 180°.

To change degrees to radians, multiply by π/180; to go back, multiply by 180/π. So 45° = 45 × π/180 = π/4, 30° = π/6 and 90° = π/2, while 2π/3 radians is 2π/3 × 180/π = 120°. One radian is 180/π ≈ 57.2958°. Radians make arc length simple: an angle of θ radians cuts an arc of length s = rθ. On a circle of radius 5 cm, 2 radians cut an arc of 10 cm. For degrees you must convert first: 60° on a circle of radius 10 cm is π/3 radians, giving an arc of 10π/3 ≈ 10.47 cm.

The unit circle

A right triangle only has angles up to 90°, but rotations go further. The unit circle removes the limit. It is the circle of radius 1 centred at the origin. Start at the point (1, 0) and turn counterclockwise by an angle t. Precalculus defines the cosine of t as the x-coordinate of the point you reach and the sine of t as its y-coordinate: (x, y) = (cos t, sin t). For angles between 0° and 90° this agrees with the triangle: the radius is a hypotenuse of length 1, so the horizontal leg is cos t and the vertical leg is sin t.

Past 90° the coordinates take signs. At 150° the point is up and to the left, so cos 150° = −√3/2 and sin 150° = 1/2. At 180° the point is (−1, 0). In the third quadrant both coordinates are negative, so sin 210° = −1/2. Tangent is y/x, the slope of the radius, and it does not exist at 90° and 270°, where x = 0.

cos² + sin² = 1

Every point on the unit circle is at distance 1 from the origin, so x² + y² = 1. Substituting x = cos t and y = sin t gives cos²t + sin²t = 1, which Precalculus calls the Pythagorean Identity. It lets you find one value from the other. If sin t = 3/5 and the angle is in the second quadrant, cos²t = 1 − 9/25 = 16/25, and because x is negative there, cos t = −4/5.

Why bother with radians at all? Because they make formulas clean. The arc length s = rθ has no stray factor of π/180, and in calculus the derivative of sin x is cos x only when x is in radians. For that reason most science and every calculus course work in radians, and degrees stay for everyday angles such as compass bearings and roof pitches.