from the (historically later) general functional notation in which However, the exponent is commonly used to denote the inverse function, not the reciprocal. For example and denote the inverse trigonometric function alternatively written The equation implies not In this case, the superscript could be considered as denoting a composed or iterated function, but negative superscripts other than are not in common use. Right-angled triangle definitions If the acute angle is given, then any right triangles that have an angle of are similar to each other. This means that the ratio of any two side lengths depends only on . Thus these six ratios define six functions of , which are the trigonometric functions. In the following definitions, the hypotenuse is the length of the side opposite the right angle, opposite represents the side opposite the given angle , and adjacent represents the side between the angle and the right angle. In a right-angled triangle, the sum of the two acute angles is a right angle, that is, or . Therefore and represent the same ratio, and thus are equal. This identity and analogous relationships between the other trigonometric functions are summarized in the following table. Radians versus degrees In geometric applications, the argument of a trigonometric function is generally the measure of an angle. For this purpose, any angular unit is convenient, and angles are most commonly measured in conventional units of degrees in which a right angle is 90° and a complete turn is 360° (particularly in elementary mathematics). However, in calculus and mathematical analysis, the trigonometric functions are generally regarded more abstractly as functions of real or complex numbers, rather than angles. In fact, the functions sin and cos can be defined for all complex numbers in terms of the exponential function via power series or as solutions to differential equations given particular initial values (see below), without reference to any geometric notions. The other four trigonometric functions (tan, cot, sec, csc) can be defined as quotients and reciprocals of sin and cos, except where zero occurs in the denominator. It can be proved, for real arguments, that these definitions coincide with elementary geometric definitions if the argument is regarded as an angle given in radians. Moreover, these definitions result in simple expressions for the derivatives and indefinite integrals for the trigonometric functions. Thus, in settings beyond elementary geometry, radians are regarded as the mathematically natural unit for