Math Problem Statement

solve please

Plot the complex number -3 + i√5. Then write it in polar form.

Solution

Let's solve the problem step by step.

We are tasked with:

  1. Plotting the complex number −3+i5-3 + i\sqrt{5} on the graph.
  2. Writing the number in polar form.

Step 1: Plot the Complex Number

The complex number is −3+i5-3 + i\sqrt{5}.

  • The real part is −3-3, so it will be located on the negative side of the real axis.
  • The imaginary part is 5\sqrt{5}, which is approximately 2.242.24. This places the point above the real axis.

Thus, the point is in the second quadrant. Checking the graphs provided, the correct plot corresponds to option B.

Step 2: Convert to Polar Form

The polar form of a complex number is expressed as: r(cos⁡θ+isin⁡θ)r (\cos \theta + i \sin \theta) where:

  • r=x2+y2r = \sqrt{x^2 + y^2} is the modulus (distance from the origin),
  • θ=tan⁡−1(yx)\theta = \tan^{-1} \left( \frac{y}{x} \right) is the argument (angle).

For −3+i5-3 + i\sqrt{5}, we have:

  • x=−3x = -3
  • y=5y = \sqrt{5}

Step 2.1: Calculate rr:

= \sqrt{9 + 5} = \sqrt{14}$$ **Step 2.2: Calculate $$\theta$$:** $$\theta = \tan^{-1} \left( \frac{y}{x} \right) = \tan^{-1} \left( \frac{\sqrt{5}}{-3} \right)$$ Since the point is in the second quadrant, the angle must satisfy: $$\theta = 180^\circ + \tan^{-1} \left( \frac{\sqrt{5}}{3} \right)$$ Using a calculator: $$\tan^{-1} \left( \frac{\sqrt{5}}{3} \right) \approx 33.7^\circ$$ Thus: $$\theta = 180^\circ - 33.7^\circ = 146.3^\circ$$ --- #### Final Answer: The polar form of $$-3 + i\sqrt{5}$$ is: $$\sqrt{14} (\cos 146.3^\circ + i \sin 146.3^\circ)$$ From the options: - **Correct choice: C** --- ### Summary: 1. The graph corresponds to **B**. 2. The polar form is: $$\sqrt{14} (\cos 146.3^\circ + i \sin 146.3^\circ)$$ Do you want a detailed explanation of any step, or have further questions? 😊 --- #### Related Questions: 1. What is the relationship between rectangular and polar forms of complex numbers? 2. How do you find the modulus of a complex number geometrically? 3. What happens to the angle when a complex number lies in different quadrants? 4. Can you convert polar form back to rectangular form? 5. What are common errors in calculating the argument of a complex number? #### Tip: When calculating the angle $$\theta$$, always consider the quadrant of the complex number to adjust the angle appropriately.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Complex Numbers
Polar Form
Trigonometry

Formulas

r = √(x² + y²)
θ = tan⁻¹(y/x)
Polar form: r(cos θ + i sin θ)

Theorems

Pythagorean theorem for modulus calculation
Inverse tangent function for angle calculation

Suitable Grade Level

Grades 10-12

Related Recommendation

Convert Complex Number 5 - 5i to Trigonometric Form

Trigonometric Form of Complex Number 3 + i

Convert Complex Number 5 - 5i to Polar Form

Expressing Complex Number 3√3 + i in Polar Form

Convert Complex Number to Polar Form: (5√3)/4 - 5/4 i