Trigonometric functions of α and −α

The points M₁ and M₂ obtained by rotating P(1, 0) through α and −α are symmetric about the Ox axis: they have the same abscissa and ordinates that differ only in sign. Therefore cos(−α) = cos α and sin(−α) = −sin α. From the definitions of tangent and cotangent, tan(−α) = −tan α and cot(−α) = −cot α. So y = cos x is an even function, while y = sin x, y = tan x and y = cot x are odd functions (see § 11). These formulas let us reduce negative angles to positive ones, for example sin(−π/6) = −sin(π/6) = −1/2. We also use them to simplify expressions: first take the minus sign out of the angle, then apply the relations of § 21.