Solving Complex Number Proof: w^2 + (5/w) - 2 = 0

[ahoy hoy]

Homework Statement

w=cos(theta) + isin(theta) where 0<theta<pi
if the complex number w^2 + (5/w) -2 is purely imaginary, show that 2cos^2 x + 5 cos (theta) -3=0.
Hence, find w.

Homework Equations

cos^2(theta) + sin^2 (theta) = 1

The Attempt at a Solution

im guessing to substitue w into the complex number.

 

[HallsofIvy]

Do you know that [itex]w^2= cos(2\theta)+ i sin(2\theta)[/itex] and that [itex]1/w= w^{-1}= cos(-\theta)+ i sin(-\theta)= cos(\theta)- i sin(\theta)[/itex]? Whoever assigned this problem certainly expects you to know that!

 

[Architeuthis Dux]

Yes, that is correct. To solve this complex number proof, you will need to substitute w=cos(theta) + isin(theta) into the given complex number. This will result in a purely imaginary number, which can be written as bi, where b is a real number. Then, equate the real and imaginary parts of the complex number to 0, and solve for theta. Once you have theta, you can plug it back into w=cos(theta) + isin(theta) to find the value of w.

Once you have found w, you can use the trigonometric identity cos^2(theta) + sin^2(theta) = 1 to simplify the expression 2cos^2(theta) + 5cos(theta) -3 into a form that only involves the variable w. This will help you to further understand the relationship between the given complex number and the final expression.

In summary, to solve this complex number proof, you will need to use substitution, trigonometric identities, and algebraic manipulation to find the value of w and simplify the expression. It is important to understand the properties of complex numbers and trigonometric functions in order to successfully solve this problem.