Cook Thornton
09/06/2024 · Middle School
The rectangular coordinates of a point are given. Find polar coordinates for the point. \( (5,5 \sqrt{3}) \)
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
To convert the rectangular coordinates \((x, y) = (5, 5\sqrt{3})\) to polar coordinates \((r, \theta)\), we use the following formulas:
1. The radius \(r\) is given by:
\[
r = \sqrt{x^2 + y^2}
\]
2. The angle \(\theta\) is given by:
\[
\theta = \tan^{-1}\left(\frac{y}{x}\right)
\]
### Step 1: Calculate \(r\)
First, we calculate \(r\):
\[
r = \sqrt{5^2 + (5\sqrt{3})^2}
\]
\[
= \sqrt{25 + 75}
\]
\[
= \sqrt{100}
\]
\[
= 10
\]
### Step 2: Calculate \(\theta\)
Next, we calculate \(\theta\):
\[
\theta = \tan^{-1}\left(\frac{y}{x}\right) = \tan^{-1}\left(\frac{5\sqrt{3}}{5}\right) = \tan^{-1}(\sqrt{3})
\]
The angle \(\tan^{-1}(\sqrt{3})\) corresponds to:
\[
\theta = \frac{\pi}{3} \quad \text{(in radians)}
\]
### Conclusion
Thus, the polar coordinates of the point \((5, 5\sqrt{3})\) are:
\[
\boxed{(10, \frac{\pi}{3})}
\]
Quick Answer
The polar coordinates of the point \((5, 5\sqrt{3})\) are \((10, \frac{\pi}{3})\).
Answered by UpStudy AI and reviewed by a Professional Tutor
Like
(0)
Ask Real Tutor
UpStudy ThothAI
Self-Developed and Ever-Improving
Thoth AI product is constantly being upgraded and optimized.
Covers All Major Subjects
Capable of handling homework in math, chemistry, biology, physics, and more.
Instant and Accurate
Provides immediate and precise solutions and guidance.
Try Now
PREV 
Ask Tutors
Ask AI
10x
Fastest way to Get Answers & Solutions
By text
Enter your question here…
By image
Re-Upload
Uploaded Files
xxxx.png0%
Submit
