Problem 40 - Algebra and Trigonometry

Question

Convert the rectangular coordinates to polar coordinates with \(r>0\) and \(0 \leq \theta<2 \pi .\) $$ (1,-2) $$

Step-by-Step Solution

Answer

The polar coordinates are \( \left(\sqrt{5}, 5.176\right) \).

Step 1: Calculate the Radius (r)

The radius, \( r \), in polar coordinates is calculated using the formula \( r = \sqrt{x^2 + y^2} \). For the point \((1, -2)\), substitute \( x = 1 \) and \( y = -2 \): \[ r = \sqrt{1^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5} \] So, \( r = \sqrt{5} \).

Step 2: Determine the Angle (θ)

The angle \( \theta \) is calculated using the formula \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \). For \( (1, -2) \), substitute \( x = 1 \) and \( y = -2 \): \[ \theta = \tan^{-1}\left(\frac{-2}{1}\right) = \tan^{-1}(-2) \] The resulting angle would be in the fourth quadrant since \( x \) is positive and \( y \) is negative. Express \( \theta \) in the range \([0, 2\pi)\): \[ \theta = 2\pi + \tan^{-1}(-2) = 2\pi - \theta' \] Where \( \theta' \) is the positive equivalent of \( \tan^{-1}(-2) \) which is approximately \( 1.107 \). Thus, \[ \theta = 2\pi - 1.107 \approx 5.176 \]

Step 3: Write the Polar Coordinates

Combine the radius and the angle to express the polar coordinates. From Step 1, \( r = \sqrt{5} \), and from Step 2, \( \theta \approx 5.176 \). Therefore, the polar coordinates of \((1, -2)\) are: \( \left(\sqrt{5}, 5.176\right) \).

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, describe a point in a plane using two values: an x-coordinate and a y-coordinate. This system represents position using straight horizontal and vertical distances from a reference point, usually called the origin.

• **X-coordinate**: Represents the horizontal distance from the origin.
• **Y-coordinate**: Represents the vertical distance from the origin.

A point in this coordinate system can be visualized as the intersection of a horizontal line from the x-value and a vertical line from the y-value. For example, the point (1, -2) means 1 unit to the right of the origin and 2 units down.

Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles, particularly right triangles.

It involves the study of functions like sine, cosine, and tangent, which relate angles in right triangles to ratios of the triangle's sides.

• **Sine (sin)**: The ratio of the opposite side to the hypotenuse.
• **Cosine (cos)**: The ratio of the adjacent side to the hypotenuse.
• **Tangent (tan)**: The ratio of the opposite side to the adjacent side.

In the context of converting rectangular coordinates to polar coordinates, we often use the tangent function. By applying the formula \( \theta = \tan^{-1}(\frac{y}{x}) \), we calculate the angle which helps us express a point in terms of polar coordinates.

Angle Calculation

Calculating angles accurately is crucial when working with polar coordinates, as it helps in properly identifying the direction of the point from the origin. The calculated angle is frequently expressed in radians, where the complete rotation from 0 to 2π represents a full circle.

To find the angle θ in radians using a point's rectangular coordinates • **Use the tangent formula**: \( \theta = \tan^{-1}(\frac{y}{x}) \).
• **Quadrant consideration**: Identify which quadrant the point is in to adjust the calculated angle as necessary.
For example, if the x-value is positive and y-value is negative, the point lies in the fourth quadrant. In such cases, the angle should be adjusted, often by adding 2π if dealing with negative angles to fit the interval [0, 2π).

This process ensures that the angle accurately represents the direction of the point in its polar form.