Question

Writing a Complex Number in Standard Form: Write the complex number in standard form a+bi

z=√8[cos(-π/3)+isin(-π/3)]

Answer 1

To write the given complex number in standard form, we use Euler's formula to rewrite it in exponential form, and then simplify the trigonometric functions to obtain the standard form.

Step-by-step explanation:

To write the given complex number in standard form, we first rewrite it using Euler's formula: √8[cos(-π/3) + isin(-π/3)] = √8e^(-iπ/3). Now, using the property e^(ix) = cos(x) + isin(x), we can express the complex number as √8(cos(-π/3) + isin(-π/3)) = √8e^(-iπ/3) = √8e^(-i(π/3)) = √8e^(i(-π/3 + 2π)) = √8e^(i(5π/3)).

Since the standard form of a complex number is a + bi, where a and b are real numbers, we can rewrite √8e^(i(5π/3)) as (√8cos(5π/3)) + i(√8sin(5π/3)).

Finally, simplifying the trigonometric functions, we have (√8)(-1/2) + i(√8)(-√3/2) = -√2 - i√6.