How to calculate the sine of a complex number?

To calculate the sine of a complex number, use the formula sin(z) = (e^(iz) - e^(-iz))/2i.

The sine of a complex number can be found using the formula sin(z) = (e^(iz) - e^(-iz))/2i, where z is a complex number and i is the imaginary unit. This formula can be derived using the exponential form of a complex number, z = x + iy = r(cosθ + isinθ), where r is the modulus and θ is the argument. To deepen your understanding of the trigonometric form, you can review it on the Trigonometric Form of Complex Numbers page.

To apply the formula, first express the complex number z in terms of its real and imaginary parts, z = x + iy. Then, substitute this expression into the formula sin(z) = (e^(iz) - e^(-iz))/2i and simplify using the properties of exponents and trigonometric functions. It's beneficial to refer to the Trigonometric Identities and Trigonometric Ratios to help simplify and understand the trigonometric calculations involved.

For example, to find the sine of the complex number z = 2 + 3i, we have:

sin(z) = (e^(i(2+3i)) - e^(-i(2+3i)))/2i
= (e^(2i-3)) - (e^(-2i+3))/2i
= (cos(2) + i sin(2))(e^(-3i)) - (cos(2) - i sin(2))(e^(3i))/2i
= (cos(2)e^(-3i) - cos(2)e^(3i))/2i + (sin(2)e^(-3i) + sin(2)e^(3i))/2
= sin(2)/2 + (cos(2)/2)i

Therefore, the sine of the complex number z = 2 + 3i is sin(z) = sin(2)/2 + (cos(2)/2)i.

A-Level Maths Tutor Summary: To calculate the sine of a complex number, use the formula sin(z) = (e^(iz) - e^(-iz))/2i. First, break down the complex number into real and imaginary parts. Then, apply the formula and simplify the expression using exponent and trigonometry rules. For instance, the sine of z = 2 + 3i is sin(z) = sin(2)/2 + (cos(2)/2)i, combining both sine and cosine components.