This online Trigonometric Functions Calculator computes trigonometric functions of a complex number (variable). When typing the imaginary part of a complex number in the appropriate field of the calculator, make sure that the symbol ‘i‘, representing the imaginary unit, is adjacent to the numeric part without space.
Trigonometric functions of one complex variable
This online calculator computes the following trigonometric functions of a complex variable \(z=x+yi\), where \(x\) and \(y\) are real numbers.
• sin() – sine function
The sine function of \(z\) is defined by:
$$ sin(z) = \sum_{n=0}^\infty \frac{(-1)^{n}}{(2n+1)!}z^{2n+1} = \frac{{e}^{iz}-{e}^{-iz}}{2i}.$$
• cos() – cosine function
The definition of the cosine function can be extended to complex argument \(z\) using the following expression:
$$ cos(z) = \frac{e^{i z} + e^{-i z}}{2}.$$
• tan() – tangent function
The tangent function (sometimes also denoted \(tg()\)) is defined in the following way:
$$tan(z)=\frac { sin(z) }{ cos(z) } =\frac { { e }^{ iz }-{ e }^{ -iz } }{ i({ e }^{ iz }+{ e }^{ -iz }) } .$$
• csc() – cosecant function
The cosecant function is defined in the following way:
$$ csc(z)=\frac { 1 }{ sin(z) }.$$
• sec() – secant function
The secant function is defined in the following way:
$$ sec(z)=\frac { 1 }{ cos(z) }.$$
• cot() – cotangent function
The cotangent function is defined in the following way:
$$ cot(z)=\frac { 1 }{ tan(z) }.$$
The inverse trigonometric functions are the inverse functions of the trigonometric functions. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, cosecant and secant functions. These functions can be expressed using complex logarithms. This extends their domains to the complex plane in a natural fashion. The following identities for principal values of the functions hold everywhere that they are defined, even on their branch cuts.
$$arcsin(z) = -i \ln \left( \sqrt{1-z^2} + iz \right) = i \ln \left( \sqrt{1-z^2} – iz \right),$$
$$arccos(z) = -i \ln \left( i \sqrt{1-z^2} + z \right) = \frac{\pi}{2} – arcsin(z),$$
$$arctan(z) = -\frac{i}{2}\ln \left(\frac{i – z}{i + z}\right) = -\frac{i}{2}\ln \left(\frac{1 + iz}{1 – iz}\right),$$
$$arccot(z) = -\frac{i}{2}\ln\left( \frac{z + i}{z – i} \right) = -\frac{i}{2}\ln\left( \frac{iz – 1}{iz + 1} \right),$$
$$arcsec(z) = -i \ln \left( i \sqrt{1 – \frac{1}{z^2}} + \frac{1}{z} \right) = \frac{\pi}{2} – arccsc(z),$$
$$arccsc(z) = -i \ln \left( \sqrt{1 – \frac{1}{z^2}} + \frac{i}{z} \right) = i \ln \left ( \sqrt{1 – \frac{1}{z^2}} – \frac{i}{z} \right).$$
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