# Reduce to x+iy: Solving for z=cos\theta+isin\theta

• kathrynag
In summary, "reducing to x+iy" means simplifying the complex number expression z=cos\theta+isin\theta into its standard form, where x and y represent the real and imaginary parts of the complex number, respectively. To solve for z=cos\theta+isin\theta, you can use the trigonometric identity cos\theta+isin\theta=e^{i\theta}. This means that z can be expressed as z=e^{i\theta}, where e is the base of the natural logarithm. You can then use this form to calculate the values of x and y, which represent the real and imaginary parts of z. Yes, you can use a calculator to solve for z=cos\
kathrynag

## Homework Statement

Reduce to x+iy
$$\frac{1+z}{1-z}$$ where z=cos$$\theta$$+isin$$\theta$$.

## The Attempt at a Solution

$$\frac{1+z}{1-z}$$
Multiply by conjugate
$$\frac{1+2z+z^{2}}{1-z^{2}}$$
When I plug in the z value nothing seems to cancel out.

You seem to have multiplied both the numerator and denominator by $(1+z)$...but that isn't really the complex conjugate of $(1-z)$ is it?...Don't you actually want to multiply by $(1-\bar{z})$ instead?

So i should multiply by 1+(costheta+isintheta)?

No, $1-z=(1-\cos\theta)-i\sin\theta$, so $\overline{1-z}=$___?

## 1. What is meant by "reducing to x+iy" in the context of solving for z=cos\theta+isin\theta?

"Reducing to x+iy" means simplifying the complex number expression z=cos\theta+isin\theta into its standard form, where x and y represent the real and imaginary parts of the complex number, respectively.

## 2. How do I solve for z=cos\theta+isin\theta?

To solve for z=cos\theta+isin\theta, you can use the trigonometric identity cos\theta+isin\theta=e^{i\theta}. This means that z can be expressed as z=e^{i\theta}, where e is the base of the natural logarithm. You can then use this form to calculate the values of x and y, which represent the real and imaginary parts of z.

## 3. Can I use a calculator to solve for z=cos\theta+isin\theta?

Yes, you can use a calculator to solve for z=cos\theta+isin\theta. Most scientific calculators have a complex number mode that allows you to enter the expression and calculate the values of x and y.

## 4. What is the significance of the complex number z=cos\theta+isin\theta?

The complex number z=cos\theta+isin\theta is significant in mathematics and physics, as it represents a point on the complex plane and can be used to perform operations such as addition, subtraction, multiplication, and division. It also has applications in solving differential equations and in electrical engineering.

## 5. Are there any tips for simplifying complex numbers like z=cos\theta+isin\theta?

One tip for simplifying complex numbers like z=cos\theta+isin\theta is to convert them into polar form, where z=r\left(cos\theta+isin\theta\right), and use the trigonometric identities to calculate the values of r, the magnitude of z, and \theta, the angle of z in the complex plane. This can make the process of solving for x and y simpler and more intuitive.

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