Complex Number: Convert 7(cos 135° + I Sin 135°) To A+bi

by Jhon Lennon 57 views

Alright, guys! Let's dive into converting a complex number from its trigonometric form to its standard form. We're going to take the complex number 7(cos 135° + i sin 135°) and rewrite it in the a + bi format. This involves understanding the relationship between trigonometric functions and complex numbers, and a little bit of coordinate geometry. So, buckle up and let's get started!

Understanding the Trigonometric Form of Complex Numbers

First, let's break down what the trigonometric form of a complex number actually means. A complex number in trigonometric or polar form is generally represented as r(cos θ + i sin θ), where:

  • r is the magnitude (or modulus) of the complex number, representing its distance from the origin in the complex plane.
  • θ (theta) is the argument of the complex number, representing the angle formed with the positive real axis.

In our case, we have 7(cos 135° + i sin 135°). This tells us that:

  • The magnitude r is 7.
  • The argument θ is 135°.

So, what does this tell us? It means that if we were to plot this complex number on the complex plane, it would be 7 units away from the origin, and the line connecting the origin to this point would make an angle of 135° with the positive real axis. Understanding this geometrical interpretation is crucial for converting it to the standard form a + bi.

Converting to Standard Form: a + bi

Now, let's convert our complex number 7(cos 135° + i sin 135°) into the standard form a + bi. To do this, we need to find the values of cos 135° and sin 135°. Recall that 135° is in the second quadrant. In the second quadrant, cosine is negative, and sine is positive.

  • cos 135° = -cos (180° - 135°) = -cos 45° = -√2 / 2
  • sin 135° = sin (180° - 135°) = sin 45° = √2 / 2

Now, we substitute these values back into our complex number:

7(cos 135° + i sin 135°) = 7((-√2 / 2) + i(√2 / 2)).

Next, distribute the 7:

7((-√2 / 2) + i(√2 / 2)) = -7√2 / 2 + i(7√2 / 2).

So, our complex number in standard form a + bi is:

-7√2 / 2 + (7√2 / 2)i

Here, a = -7√2 / 2 and b = 7√2 / 2. That's it! We've successfully converted the complex number from its trigonometric form to standard form.

Step-by-Step Breakdown

Let's summarize the steps we took to convert the complex number 7(cos 135° + i sin 135°) to the standard form a + bi:

  1. Identify the magnitude and argument:
    • Magnitude r = 7
    • Argument θ = 135°
  2. Find the values of cos θ and sin θ:
    • cos 135° = -√2 / 2
    • sin 135° = √2 / 2
  3. Substitute the values into the trigonometric form:
    • 7(cos 135° + i sin 135°) = 7((-√2 / 2) + i(√2 / 2))
  4. Distribute the magnitude:
    • 7((-√2 / 2) + i(√2 / 2)) = -7√2 / 2 + i(7√2 / 2)
  5. Write the complex number in standard form:
    • -7√2 / 2 + (7√2 / 2)i

Following these steps makes the conversion process straightforward and easy to understand. Remember, the key is to correctly evaluate the cosine and sine functions for the given angle and then apply the distributive property.

Common Mistakes to Avoid

When converting complex numbers from trigonometric to standard form, there are a few common mistakes you should watch out for:

  • Incorrectly evaluating trigonometric functions:
    • Make sure you know the correct signs for sine and cosine in each quadrant. For example, cosine is negative in the second and third quadrants, while sine is negative in the third and fourth quadrants. Remembering the acronym "All Students Take Calculus" (ASTC) can help: All (quadrant I), Sine (quadrant II), Tangent (quadrant III), Cosine (quadrant IV) are positive in those quadrants.
  • Forgetting to distribute the magnitude:
    • After substituting the values of cos θ and sin θ, remember to multiply both the real and imaginary parts by the magnitude r.
  • Mixing up sine and cosine:
    • Ensure you correctly assign the cosine and sine values to the real and imaginary parts, respectively. The real part corresponds to cos θ, and the imaginary part corresponds to sin θ.
  • Not simplifying radicals:
    • Always simplify radicals when possible. For example, √4 should be simplified to 2.

By being mindful of these potential pitfalls, you can avoid errors and ensure accurate conversions.

Why This Conversion is Important

Converting complex numbers between trigonometric and standard forms is not just an academic exercise; it has practical applications in various fields, including:

  • Electrical Engineering:
    • Complex numbers are used extensively in AC circuit analysis. Converting between forms helps in simplifying calculations involving impedance, voltage, and current.
  • Physics:
    • In quantum mechanics, complex numbers are used to describe wave functions. Converting between forms can simplify calculations involving wave interference and superposition.
  • Mathematics:
    • Complex numbers are fundamental in many areas of mathematics, including calculus, algebra, and geometry. Understanding their different forms is crucial for solving complex problems.
  • Computer Graphics:
    • Complex numbers can be used to represent rotations and scaling in 2D graphics. Converting between forms can simplify transformations and animations.

Understanding how to convert between trigonometric and standard forms of complex numbers allows for easier manipulation and application in these diverse fields. Whether you're analyzing circuits, studying quantum mechanics, or creating computer graphics, this skill can be incredibly valuable.

Practice Problems

To solidify your understanding, let's work through a few practice problems:

  1. Convert the complex number 5(cos 225° + i sin 225°) to standard form.
  2. Convert the complex number 2(cos 60° + i sin 60°) to standard form.
  3. Convert the complex number 4(cos 300° + i sin 300°) to standard form.

Solutions

  1. 5(cos 225° + i sin 225°):
    • cos 225° = -√2 / 2
    • sin 225° = -√2 / 2
    • 5((-√2 / 2) + i(-√2 / 2)) = -5√2 / 2 - (5√2 / 2)i
  2. 2(cos 60° + i sin 60°):
    • cos 60° = 1 / 2
    • sin 60° = √3 / 2
    • 2((1 / 2) + i(√3 / 2)) = 1 + i√3
  3. 4(cos 300° + i sin 300°):
    • cos 300° = 1 / 2
    • sin 300° = -√3 / 2
    • 4((1 / 2) + i(-√3 / 2)) = 2 - 2i√3

Working through these problems will help you become more comfortable with the conversion process and improve your accuracy. Remember to take your time, double-check your calculations, and don't be afraid to refer back to the steps we discussed earlier.

Conclusion

So, there you have it! Converting the complex number 7(cos 135° + i sin 135°) to the standard form a + bi is a straightforward process once you understand the relationship between trigonometric functions and complex numbers. By finding the values of cos 135° and sin 135°, substituting them into the expression, and simplifying, we found that 7(cos 135° + i sin 135°) = -7√2 / 2 + (7√2 / 2)i. This conversion is a fundamental skill in various fields, including engineering, physics, and mathematics, making it a valuable tool in your problem-solving arsenal. Keep practicing, and you'll become a pro in no time! Keep rocking!