Sine, cosine, tangent and cotangent of an angle

On the upper unit semicircle (centre at the origin, radius 1) take a point M(x; y); let ray OM make an angle α with ray Ox. Definition: sinα = y, cosα = x, tgα = y/x (α ≠ 90°), ctgα = x/y (α ≠ 0°, 180°). For an acute angle these agree with the usual right-triangle definitions. The Pythagorean theorem gives sin²α + cos²α = 1 (the basic trigonometric identity), tgα = sinα/cosα, tgα·ctgα = 1 and 1 + tg²α = 1/cos²α. Reduction formulas: sin(90° − α) = cosα, cos(90° − α) = sinα, sin(180° − α) = sinα, cos(180° − α) = −cosα. For an obtuse angle x < 0, so cosine, tangent and cotangent are negative while sine is positive. Ulugh Beg (1394–1449) compiled trigonometric tables for astronomical calculations at the Samarkand observatory.