i can not figure out how to write the complex number in rectangular form the question is 4(cos 5Pie/6 + i sin 5 pie/6)
Express in the form a + bi, where a and b are real numbers.
12(cos 5𝜋/4+ i sin 5𝜋/4)
we know cos 5π/6 = =√3/2 and sin 5π/6 = 1/2
so
4(cos 5Pie/6 + i sin 5 pie/6)
= 4(-√3/2 + i(1/2) )
= -2√3 + 2i
To write a complex number in rectangular form, you need to convert it from its polar form representation. In this case, the given complex number is in the form of:
z = 4(cos(5π/6) + i sin(5π/6))
To convert it to rectangular form, you can use the trigonometric identities:
cosθ = Re(z) / |z|
sinθ = Im(z) / |z|
where θ is the argument of the complex number, Re(z) is the real part, Im(z) is the imaginary part, and |z| is the modulus or magnitude of the complex number.
1. Find the modulus:
The modulus of the complex number is the absolute value of the given number:
|z| = 4
2. Find the argument:
The argument of the complex number is the angle formed between the positive x-axis and the line connecting the origin and the complex number. In this case, the argument is 5π/6.
3. Calculate Re(z) and Im(z):
Re(z) = |z| * cosθ = 4 * cos(5π/6) = -2√3
Im(z) = |z| * sinθ = 4 * sin(5π/6) = 2
4. Write the complex number in rectangular form:
z = Re(z) + i Im(z) = -2√3 + 2i
Therefore, the complex number 4(cos(5π/6) + i sin(5π/6)) can be written in rectangular form as -2√3 + 2i.