Advanced Complex Trigonometry Calculator

Calculator inputs

Real part a

Imaginary part b

Input unit

Decimal places

Output notation

Enter z = a + bi. The calculator returns circular, reciprocal, and hyperbolic functions, plus modulus, argument, conjugate, and Euler form.

Example data table

Input z	sin(z)	cos(z)	tan(z)
1 + 0i	0.841471 + 0i	0.540302 + 0i	1.557408 + 0i
0 + 1i	0 + 1.175201i	1.543081 + 0i	0 + 0.761594i
1 + 1i	1.298458 + 0.634964i	0.833730 - 0.988898i	0.271753 + 1.083923i

Formula used

Let z = x + iy

x is the real component and y is the imaginary component.

sin(x + iy)

sin(x) cosh(y) + i cos(x) sinh(y)

cos(x + iy)

cos(x) cosh(y) - i sin(x) sinh(y)

tan(z)

tan(z) = sin(z) / cos(z)

cot(z), sec(z), csc(z)

cot(z) = cos(z) / sin(z), sec(z) = 1 / cos(z), csc(z) = 1 / sin(z)

sinh(x + iy)

sinh(x) cos(y) + i cosh(x) sin(y)

cosh(x + iy)

cosh(x) cos(y) + i sinh(x) sin(y)

tanh(z)

tanh(z) = sinh(z) / cosh(z)

Modulus and argument

|z| = √(x² + y²), arg(z) = atan2(y, x)

Conjugate and Euler form

conj(z) = x - iy, z = r[cos(θ) + i sin(θ)] where r = |z| and θ = arg(z)

How to use this calculator

Enter the real part a and imaginary part b for z = a + bi.
Choose radians if the components should be evaluated directly.
Choose degrees if both entered components represent degree-based values.
Set decimal places and preferred notation for the displayed outputs.
Click the calculate button to generate the result panel.
Review circular, reciprocal, and hyperbolic values in the result table.
Check modulus, argument, conjugate, and Euler form for the same input.
Use the CSV or PDF buttons to export the computed result set.

FAQs

1. What does this calculator compute?

It evaluates a complex input z = a + bi and returns sin, cos, tan, cot, sec, csc, sinh, cosh, tanh, modulus, argument, conjugate, and Euler form in one place.

2. How are complex trigonometric values found?

The page uses closed-form identities for x + iy. Those formulas combine standard trigonometric functions with hyperbolic functions, then apply reciprocal or quotient rules where needed.

3. What is the difference between entered z and evaluation z?

Entered z is the exact number you typed. Evaluation z is the version used internally for function calculations. In degree mode, both components are converted to radians first.

4. Why can some outputs become undefined?

Reciprocal and quotient functions depend on denominators such as sin(z) or cos(z). If that denominator is zero or extremely close to zero, the related result is marked undefined.

5. What does the modulus represent?

The modulus is the distance of z from the origin on the complex plane. It equals √(a² + b²) and helps convert the number into polar or Euler form.

6. What does the argument represent?

The argument is the angle between the positive real axis and the vector representing z. This page shows that angle in both radians and degrees.

7. When should I use scientific notation?

Use scientific notation when values are very large, very small, or when you want easier comparison across outputs with different magnitudes.

8. What do the export buttons include?

Both exports capture the computed result table shown after submission. The CSV creates spreadsheet-ready rows, while the PDF produces a clean tabular report for sharing or saving.