Trigonometric forms, also known as trigonometric representations or polar forms, are alternative ways to express complex numbers. A complex number is a number that comprises a real part and an imaginary part, and it's usually denoted as "a + bi", where "a" is the real part and "b" is the imaginary part. In trigonometric form, a complex number is represented in terms of its magnitude and argument (angle).
The trigonometric form of a complex number "z" is given by:
z = r(cos(θ) + i*sin(θ))
Where:
r is the magnitude (or modulus) of the complex number, calculated as the square root of the sum of the squares of its real and imaginary parts: r = √(a^2 + b^2)
θ is the argument (angle) of the complex number, measured counterclockwise from the positive real axis in the complex plane.
This form is also known as the polar form, as it relies on polar coordinates. It represents the complex number as a vector with a certain magnitude and direction (angle) in the complex plane.
To convert a complex number from its rectangular form (a + bi) to trigonometric form (r(cos(θ) + i*sin(θ))), you can use the following steps:
Calculate the magnitude (r) using the formula: r = √(a^2 + b^2)
Calculate the argument (θ) using trigonometric functions like atan2: θ = atan2(b, a)
Write the complex number in trigonometric form using the calculated values of r and θ.
To convert a complex number from its trigonometric form back to rectangular form:
Multiply the magnitude (r) by cos(θ) to get the real part.
Multiply the magnitude (r) by sin(θ) to get the imaginary part.
Keep in mind that there's also a shorthand notation for the trigonometric form using the exponential function:
z = re^(iθ)
Where "e" is the base of the natural logarithm (approximately 2.71828) and "i" is the imaginary unit.
This form is particularly useful when performing operations like multiplication, division, and exponentiation of complex numbers, as these operations become simpler in trigonometric or exponential forms compared to the rectangular form.