Trigonometry (11th Edition)
11th Edition
ISBN: 9780134217437
Author: Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher: PEARSON
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Transcribed Image Text:**Educational Website Content: Letter Matching Exercise**
**Objective:**
Match each lowercase letter with the corresponding uppercase letter.
**Instructions:**
Look at each lowercase letter in the left-hand column and find its matching uppercase letter in the right-hand column. Practice matching them correctly to reinforce your knowledge of the alphabet.
**Matching Pairs:**
- a -> P
- b -> Q
- c -> R
- d -> S
This exercise helps in identifying and learning the uppercase counterparts of the lowercase letters.
![**Understanding the Complex Plane:**
The complex plane, also known as the Argand plane, is a way to visualize and work with complex numbers. The horizontal axis (Real axis) represents the real part of a complex number, while the vertical axis (Imaginary axis) represents the imaginary part.
**Graph Description:**
The graph represents several points on the complex plane:
1. **Point S:** Located at (-9, 5), where -9 is the real part and 5 is the imaginary part.
2. **Point P:** Located at (4, 5), where 4 is the real part and 5 is the imaginary part.
3. **Point R:** Located at (-12, 0), where -12 is the real part and 0 is the imaginary part.
4. **Point Q:** Located at (-7, -10), where -7 is the real part and -10 is the imaginary part.
5. **Point z1:** Located at (-3, -5), where -3 is the real part and -5 is the imaginary part.
6. **Point z2:** Located at (2, -6), where 2 is the real part and -6 is the imaginary part.
**Problem:**
Given the points z1 and z2 on the complex plane, identify which point represents the value of \(z_1 + z_2^2\).
### Solving the Problem:
1. **Points Given:**
- \(z_1 = -3 - 5i\)
- \(z_2 = 2 - 6i\)
2. **Calculating \(z_2^2\):**
\[
z_2^2 = (2 - 6i)^2 = 2^2 - 2 \cdot 2 \cdot 6i + (-6i)^2 = 4 - 24i - 36 = -32 - 24i
\]
3. **Calculating \(z_1 + z_2^2\):**
\[
z_1 + z_2^2 = (-3 - 5i) + (-32 - 24i) = -35 - 29i
\]
Therefore, we need to identify the point on the complex plane that corresponds to \((-35, -29)\).
Given the coordinates of points S, P, R,](https://content.bartleby.com/qna-images/question/6e3b5c2b-f983-44bf-8639-6b8422538878/c1a832e5-9a6d-4036-b833-008d25bb07c4/r7s8lnd_processed.png)
Transcribed Image Text:**Understanding the Complex Plane:**
The complex plane, also known as the Argand plane, is a way to visualize and work with complex numbers. The horizontal axis (Real axis) represents the real part of a complex number, while the vertical axis (Imaginary axis) represents the imaginary part.
**Graph Description:**
The graph represents several points on the complex plane:
1. **Point S:** Located at (-9, 5), where -9 is the real part and 5 is the imaginary part.
2. **Point P:** Located at (4, 5), where 4 is the real part and 5 is the imaginary part.
3. **Point R:** Located at (-12, 0), where -12 is the real part and 0 is the imaginary part.
4. **Point Q:** Located at (-7, -10), where -7 is the real part and -10 is the imaginary part.
5. **Point z1:** Located at (-3, -5), where -3 is the real part and -5 is the imaginary part.
6. **Point z2:** Located at (2, -6), where 2 is the real part and -6 is the imaginary part.
**Problem:**
Given the points z1 and z2 on the complex plane, identify which point represents the value of \(z_1 + z_2^2\).
### Solving the Problem:
1. **Points Given:**
- \(z_1 = -3 - 5i\)
- \(z_2 = 2 - 6i\)
2. **Calculating \(z_2^2\):**
\[
z_2^2 = (2 - 6i)^2 = 2^2 - 2 \cdot 2 \cdot 6i + (-6i)^2 = 4 - 24i - 36 = -32 - 24i
\]
3. **Calculating \(z_1 + z_2^2\):**
\[
z_1 + z_2^2 = (-3 - 5i) + (-32 - 24i) = -35 - 29i
\]
Therefore, we need to identify the point on the complex plane that corresponds to \((-35, -29)\).
Given the coordinates of points S, P, R,
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