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Ptolemy's theorem states that the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. When those side-lengths are expressed in terms of the sin and cos values shown in the figure above, this yields the angle sum trigonometric identity for sine: sin(α + β) = sin α cos β + cos α sin β.
satisfying respectively y(0) = 0, y ′ (0) = 1 and y(0) = 1, y ′ (0) = 0. It follows from the theory of ordinary differential equations that the first solution, sine, has the second, cosine, as its derivative, and it follows from this that the derivative of cosine is the negative of the sine. The identity is equivalent to the assertion that ...
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 in radians. [ 5 ]
The sine and tangent small-angle approximations are used in relation to the double-slit experiment or a diffraction grating to develop simplified equations like the following, where y is the distance of a fringe from the center of maximum light intensity, m is the order of the fringe, D is the distance between the slits and projection screen ...
For example, the sine of angle θ is defined as being the length of the opposite side divided by the length of the hypotenuse. The six trigonometric functions are defined for every real number , except, for some of them, for angles that differ from 0 by a multiple of the right angle (90°).
Some of these formulas are expressed in terms of the Cartesian expansion of the spherical harmonics into polynomials in x, y, z, and r. For purposes of this table, it is useful to express the usual spherical to Cartesian transformations that relate these Cartesian components to θ {\displaystyle \theta } and φ {\displaystyle \varphi } as
The tangent half-angle substitution relates an angle to the slope of a line. Introducing a new variable = , sines and cosines can be expressed as rational functions of , and can be expressed as the product of and a rational function of , as follows: = +, = +, = +.
For example, the derivative of the sine function is written sin ′ (a) = cos(a), meaning that the rate of change of sin(x) at a particular angle x = a is given by the cosine of that angle. All derivatives of circular trigonometric functions can be found from those of sin( x ) and cos( x ) by means of the quotient rule applied to functions such ...