Problem 49 - College Trigonometry

Question

Convert the point from rectangular coordinates into polar coordinates with \(r \geq 0\) and \(0 \leq \theta<2 \pi\). $$ (-8,1) $$

Step-by-Step Solution

Answer

The polar coordinates are \((\sqrt{65}, 3.018)\).

Step 1: Understanding the Given Coordinates

The given rectangular coordinates are (-8, 1). These correspond to an x-value of -8 and a y-value of 1 on the Cartesian plane.

Step 2: Calculate the Radius

To find the radius, use the formula:\[r = \sqrt{x^2 + y^2}\]Plugging in the values:\[r = \sqrt{(-8)^2 + (1)^2} = \sqrt{64 + 1} = \sqrt{65}\]Therefore, the radius \(r = \sqrt{65}\).

Step 3: Determine the Angle θ

Use the tangent function to find the angle:\[\tan \theta = \frac{y}{x} = \frac{1}{-8} = -\frac{1}{8}\]Take the inverse tangent:\[\theta = \arctan(-\frac{1}{8})\]The calculator gives \(\theta \approx -0.124\) radians. Since the point (-8,1) is in the second quadrant, we add \(\pi\) to this result to adjust:\[\theta = -0.124 + \pi \approx 3.018\]

Step 4: Verify the Range of θ

Ensure that \(\theta\) is within the specified range of \(0 \leq \theta < 2\pi\). The calculated \(\theta \approx 3.018\) radians is in this range.Thus, the polar coordinates are correct.

Step 5: Conclusion: Writing the Polar Coordinates

The polar coordinates for the point (-8,1) are:\[(r,\theta) = (\sqrt{65}, 3.018)\]

Key Concepts

Rectangular Coordinates

Rectangular coordinates are a way to specify the position of a point on a grid, often called the Cartesian plane. In this system, each point is determined by two values:

• The x-coordinate, which is the horizontal distance from the origin.
• The y-coordinate, which is the vertical distance from the origin.

Using these two values, you can locate any point on the plane. For example, for the point given as (-8, 1), the x-value is -8. This indicates that the point is 8 units to the left of the origin (where x is 0). The y-value is 1, meaning it is 1 unit above the origin on the vertical axis. Together, they give a complete picture of the point's location in the Cartesian grid.

Radius Calculation

The radius in polar coordinates is like the distance from the origin to the point. You can think of it as the hypotenuse of a right triangle formed in a plane. To calculate this, we use the Pythagorean theorem.The formula for the radius \(r\) is:\[r = \sqrt{x^2 + y^2}\]For the point (-8, 1), plug in the x-coordinate and y-coordinate:

• \(x = -8\)
• \(y = 1\)

Then calculate:\[r = \sqrt{(-8)^2 + (1)^2} = \sqrt{64 + 1} = \sqrt{65}\]This shows you how far the point is from the center, or origin, of the plane. Thus, the radius is \(\sqrt{65}\).

Angle Determination

The angle in polar coordinates shows the direction of the point from the origin. To find this angle, you can use the tangent function, which relates the y-coordinate to the x-coordinate as follows:\[\tan \theta = \frac{y}{x}\]In our example, for the point (-8, 1):

• \(\tan \theta = \frac{1}{-8} = -\frac{1}{8}\)

To calculate \(\theta\), take the inverse tangent:\[\theta = \arctan\left(-\frac{1}{8}\right)\]With a calculator, this gives approximately \(\theta \approx -0.124\) radians. This is a critical part of converting coordinates because it defines where the point lies directionally, considering its distance in radians from the positive x-axis.

Quadrant Adjustment

In the Cartesian plane, different quadrants change how you interpret angles due to their positioning. The point (-8, 1) lands in the second quadrant, as it has a negative x and positive y value. When converting from rectangular to polar coordinates, you must adjust the angle \(\theta\) to reflect this.When the inverse tangent gives a negative angle, but the point is in the second quadrant, you should add \(\pi\) to the angle to correct it. This is because angles in the second quadrant range from \(\pi/2\) to \(\pi\).

• Initially, \(\theta \approx -0.124\) radians.
• Adjust by adding \(\pi\) for positions within the second quadrant: \(\theta = -0.124 + \pi \approx 3.018\).

This adjustment ensures the angle \(\theta\) fits within the range of \(0 \leq \theta < 2\pi\), which is the standard for polar coordinates. Thus, the adjusted angle \(\theta\) is about 3.018 radians, confirming the point's location within the correct quadrant.