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Tracing the y component of a circle while going around the circle results in a sine wave (red). Tracing the x component results in a cosine wave (blue). Both waves are sinusoids of the same frequency but different phases. A sine wave, sinusoidal wave, or sinusoid (symbol: ∿) is a periodic wave whose waveform (shape) is the trigonometric sine ...
(The cosine may be used instead of sine, depending on where one considers each period to start.) Usually, whole turns are ignored when expressing the phase; so that φ ( t ) {\displaystyle \varphi (t)} is also a periodic function, with the same period as F {\displaystyle F} , that repeatedly scans the same range of angles as t {\displaystyle t ...
In mathematics, sine and cosine are trigonometric functions of an angle.The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite that angle to the length of the longest side of the triangle (the hypotenuse), and the cosine is the ratio of the length of the adjacent leg to that of the ...
The sine and the cosine functions, for example, are used to describe simple harmonic motion, which models many natural phenomena, such as the movement of a mass attached to a spring and, for small angles, the pendular motion of a mass hanging by a string. The sine and cosine functions are one-dimensional projections of uniform circular motion.
The modern Fourier transform concisely contains both the sine and cosine transforms. Since the sine and cosine transforms use sine and cosine waves instead of complex exponentials and don't require complex numbers or negative frequency, they more closely correspond to Joseph Fourier's original transform equations and are still preferred in some ...
A triangle wave with period p and amplitude a can be expressed in terms of sine and arcsine (whose value ranges from −π/2 to π/2): = ( ()). The identity = can be used to convert from a triangle "sine" wave to a triangular "cosine" wave.
The same sinusoidal plane wave above can also be expressed in terms of sine instead of cosine using the elementary identity = (+ /) (,) = ((^) + ′) where ′ = + /.Thus the value and meaning of the phase shift depends on whether the wave is defined in terms of sine or co-sine.
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.