Math Problem Statement

(Cos x+i sin x)^4/(sin x+ i cos x)^5

Solution

The given expression is:

(cosx+isinx)4(sinx+icosx)5\frac{(\cos x + i \sin x)^4}{(\sin x + i \cos x)^5}

To simplify this expression, let's break it down step by step.

Step 1: Use Euler's formula

Euler's formula states that: eix=cosx+isinxe^{ix} = \cos x + i \sin x

So, we can rewrite the expression as: (eix)4(ei(π2x))5\frac{(e^{ix})^4}{(e^{i(\frac{\pi}{2} - x)})^5}

Step 2: Simplify the powers

ei4xei5(π2x)\frac{e^{i4x}}{e^{i5(\frac{\pi}{2} - x)}}

This simplifies to: ei(4x5(π2x))e^{i(4x - 5\left(\frac{\pi}{2} - x\right))}

Step 3: Distribute the exponent

ei(4x5π2+5x)=ei(9x5π2)e^{i(4x - \frac{5\pi}{2} + 5x)} = e^{i(9x - \frac{5\pi}{2})}

Step 4: Use Euler's formula again

Using Euler's formula in reverse, we get: cos(9x5π2)+isin(9x5π2)\cos\left(9x - \frac{5\pi}{2}\right) + i\sin\left(9x - \frac{5\pi}{2}\right)

Thus, the simplified expression is: cos(9x5π2)+isin(9x5π2)\cos\left(9x - \frac{5\pi}{2}\right) + i\sin\left(9x - \frac{5\pi}{2}\right)

Final Answer

cos(9x5π2)+isin(9x5π2)\boxed{\cos\left(9x - \frac{5\pi}{2}\right) + i\sin\left(9x - \frac{5\pi}{2}\right)}

Would you like more details or have any questions?


  1. How can Euler's formula be applied to other trigonometric expressions?
  2. What is the significance of the angle 5π2\frac{5\pi}{2} in this context?
  3. How would the expression change if the powers were different?
  4. Can you solve a similar problem involving different trigonometric functions?
  5. How does this concept relate to the complex plane?

Tip: Euler's formula is a powerful tool in complex analysis, connecting trigonometric functions with exponential functions.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Complex Numbers
Trigonometry
Euler's Formula

Formulas

Euler's formula: e^{ix} = \cos x + i \sin x

Theorems

-

Suitable Grade Level

Undergraduate