Finding the Value of (z - frac{1}{z}) Where (z costheta sintheta)
In this article, we will explore the process of determining the value of (z - frac{1}{z}) where (z costheta sintheta). This involves the use of complex numbers, trigonometric identities, and Euler's formula. The solution relies on expressing trigonometric functions in terms of complex exponentials and simplifying the expression step by step.
Expressing (z) as a Complex Number
Let's start by expressing (z costheta sintheta) in a form that allows us to manipulate it more easily. By using Euler's formula, we can rewrite (z).
Euler's Formula: (e^{itheta} costheta isintheta)
Therefore,
z costheta sintheta frac{1}{2}(sin2theta) frac{1}{2}(cos(frac{pi}{2} - 2theta))
However, using the direct form for simplification, we express (z) as:
z costheta sintheta frac{1}{2}(cos2theta - 1) frac{1}{2}(e^{i2theta} - 1)
Finding (frac{1}{z})
To find (frac{1}{z}), we can follow these steps:
[frac{1}{z} frac{1}{costheta sintheta} frac{2}{sin2theta} frac{2}{2sintheta costheta} frac{1}{sintheta costheta}]
Using the complex form:
[frac{1}{z} frac{1}{e^{i2theta}} e^{-i2theta} cos(-2theta) - isin(-2theta) cos2theta isin2theta]
Here, we used the fact that (e^{-itheta} cos(-theta) isin(-theta)).
Evaluating (z - frac{1}{z})
Now, we can compute (z - frac{1}{z}) by combining the terms:
[z - frac{1}{z} costheta sintheta - frac{1}{costheta sintheta} costheta sintheta - frac{1}{2}(cos2theta - 1)]
Breaking it down:
[z - frac{1}{z} costheta sintheta - frac{1}{2}(cos2theta - 1)]
Simplifying further:
[z - frac{1}{z} costheta sintheta - frac{1}{2}(2cos^2theta - 1 - 1)]
[ costheta sintheta - cos^2theta frac{1}{2}]
[ costheta sintheta - cos^2theta frac{1}{2}]
After simplification, we get:
[z - frac{1}{z} 2isintheta]
Thus, the value of (z - frac{1}{z}) is:
[z - frac{1}{z} 2isintheta]
Conclusion: Using the properties of complex numbers and Euler's formula, we can simplify the expression to find (z - frac{1}{z}) for any angle (theta). This method showcases the interplay between trigonometric identities and complex exponential functions.