Integral Calculators in Terms of Sine

Sine Function

• Sine is the first of three trigonometric functions. In right triangles, these functions define ratios between the shape's three sides, related to an angle (θ). Specifically, sine gives the ratio between the side opposite θ and the triangle's hypotenuse. It is often written sin(θ) and has values between -1 and 1.

Sine Cardinal Function

• Sine cardinal is a function used in several engineering projects, including signal processing. It plays a vital role in Fourier transforms and analysis. The formula shorthand for the function is sinc(x). A sine cardinal function with the x value scaled by a factor of pi is termed normalized. Sine cardinal functions without this scaling factor are termed unnormalized.

Integration of Sine Functions

• Sine is intrinsically linked to the cosine function, and calculus takes full advantage of this link. The integral of a sine is equal to the negative cosine of that angle plus a constant (C).
  The equation is as follows: ͪ7;sin(θ) dθ = -cos(θ) + C. Most calculators are capable of working out this equation.

Integration With Sine Cardinal Functions

• Sine cardinal functions are not as straightforward as sine functions. Although driven by the sine function, the sine cardinal function has a more complex definition, which is: sinc(x) = [sin(x)]/x. In the normalized version, a factor of pi scales the x-value. Therefore, the formula can be rewritten: sinc(x*pi) = [sin(x*pi)]/(x*pi). Integrating the sine cardinal function plays a key component in performing a Fourier analysis. Calculators typically offer only a good approximation of the solution to this integrated function. As the value of x increases past pi, the time required to calculate the actual integral also increases. To compensate, calculators will often offer a rational approximation of the integral instead of performing the actual integral.