Problem 6 - Calculus

Question

Find the polar coordinates \(r \geqslant 0\) and \(0 \leqslant 0<2 \pi\) of these points. $$ (x, y)=(3,4) $$

Step-by-Step Solution

Answer

The polar coordinates are \((5, 0.927)\).

Step 1: Calculate the Radius

To find the polar coordinates, we first need to calculate the radius \(r\), which is the distance from the origin \((0,0)\) to the point \((x,y)\). Use the formula \( r = \sqrt{x^2 + y^2} \). For the point \((3, 4)\), substitute \(x = 3\) and \(y = 4\):\[ r = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5.\] This means \( r = 5 \).

Step 2: Determine the Angle

Next, find the angle \(\theta\) using the formula \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \). Here, \(x = 3\) and \(y = 4\), so:\[ \theta = \tan^{-1}\left(\frac{4}{3}\right). \] Calculating \( \theta \) gives us \(\theta \approx 0.927\) radians, or approximately \(53.13\) degrees.

Step 3: Confirm the Quadrant

Since both \(x\) and \(y\) are positive, the point \((3,4)\) is in the first quadrant where angles range from \(0\) to \(\frac{\pi}{2}\). Therefore, \(\theta = 0.927\) falls correctly in this range. Thus, the angle \(\theta\) is accurate for the point's quadrant.

Step 4: Express the Polar Coordinates

Combine the radius and angle to express the polar coordinates of the point. The polar coordinates \((r, \theta)\) are \((5, 0.927)\). Make sure \(\theta\) is in radians and within the stipulated range \(0 \leq \theta < 2\pi\).

Key Concepts

Radius Calculation

In polar coordinates, the radius represents the distance from the origin \((0,0)\) to the given point \((x,y)\). This is crucial because it tells us how far we need to "shoot" out from the origin to reach the point. To calculate it, we use the formula for the radius:

• \( r = \sqrt{x^2 + y^2} \)

For instance, if you have a point like \((3, 4)\):

• Square both components: \(3^2 = 9\) and \(4^2 = 16\).
• Add them together: \(9 + 16 = 25\).
• Find the square root of the sum: \(\sqrt{25} = 5\).

Therefore, the radius \(r\) is \(5\). This helps in converting any point from Cartesian form to polar form by giving us one of the two main components needed, the length of the radius.

Angle Determination

The angle \(\theta\) in polar coordinates shows us the direction in which the point is located relative to the positive x-axis. This angle is crucial because it tells you where to "look" after moving out the radius distance. To determine this angle, use the arctangent function:

• \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \)

With the point \((3, 4)\):

• The calculation becomes \( \theta = \tan^{-1}\left(\frac{4}{3}\right) \).

This yields \(\theta \approx 0.927\) radians.
Converting to degrees, this angle is approximately \(53.13^\circ\). It's important to always check that \(\theta\) is within the appropriate range for polar coordinates, \(0 \leq \theta < 2\pi\), to ensure correctness.

Quadrants in Polar Coordinates

Understanding quadrants helps clarify where a point is situated based on the signs of its Cartesian coordinates \((x,y)\). Knowing the quadrant is especially beneficial in confirming whether your calculated angle \(\theta\) aligns correctly with the point's position. Polar coordinates divide the plane into four quadrants based on the angle’s range:- **First Quadrant:** \(0 \leq \theta < \frac{\pi}{2}\) — both \(x\) and \(y\) are positive. - **Second Quadrant:** \(\frac{\pi}{2} \leq \theta < \pi\) — \(x\) is negative and \(y\) is positive.- **Third Quadrant:** \(\pi \leq \theta < \frac{3\pi}{2}\) — both \(x\) and \(y\) are negative.- **Fourth Quadrant:** \(\frac{3\pi}{2} \leq \theta < 2\pi\) — \(x\) is positive and \(y\) is negative.Looking at the point \((3, 4)\), since both components are positive, it lies in the first quadrant. This confirms our angle \(\theta \approx 0.927\) radians, as it falls between \(0\) and \(\frac{\pi}{2}\), verifying its accuracy. Such confirmation ensures that our polar representation \((5, 0.927)\) beautifully matches the original Cartesian coordinates.