Problem 17 - College Trigonometry

Question

Convert the point from polar coordinates into rectangular coordinates. $$ \left(5, \frac{7 \pi}{4}\right) $$

Step-by-Step Solution

Answer

The rectangular coordinates are \(\left(\frac{5\sqrt{2}}{2}, -\frac{5\sqrt{2}}{2}\right)\).

Step 1: Recognize Polar Coordinates Structure

In polar coordinates, a point is represented as \((r, \theta)\), where \(r\) is the distance from the origin and \(\theta\) is the angle from the positive x-axis. Here, \(r = 5\) and \(\theta = \frac{7\pi}{4}\).

Step 2: Convert to Rectangular Coordinates

Rectangular coordinates are given as \((x, y)\). To convert, use the formulas: \[ x = r \cos(\theta) \]\[ y = r \sin(\theta) \]

Step 3: Calculate x-coordinate

Using the formula for \(x\):\[ x = 5 \cos\left(\frac{7\pi}{4}\right) \]Recall that \(\frac{7\pi}{4}\) is equivalent to 315° or -45°, where cosine is positive. Thus, \(\cos\left(\frac{7\pi}{4}\right) = \frac{\sqrt{2}}{2}\).So, \[ x = 5 \times \frac{\sqrt{2}}{2} = \frac{5\sqrt{2}}{2} \]

Step 4: Calculate y-coordinate

Using the formula for \(y\):\[ y = 5 \sin\left(\frac{7\pi}{4}\right) \]At \(\frac{7\pi}{4}\) (315° or -45°), the sine function is negative. Thus, \(\sin\left(\frac{7\pi}{4}\right) = -\frac{\sqrt{2}}{2}\).So, \[ y = 5 \times -\frac{\sqrt{2}}{2} = -\frac{5\sqrt{2}}{2} \]

Step 5: Present the Solution

Thus, the rectangular coordinates (x, y) for the point are:\[ \left(\frac{5\sqrt{2}}{2}, -\frac{5\sqrt{2}}{2}\right) \]

Key Concepts

Polar Coordinates

In the coordinate system known as polar coordinates, a point in the plane is defined uniquely in terms of the distance from a fixed reference point and the angle relative to a fixed direction. The fixed point is called the origin, similar to the center of a circle, and the angle is measured from a reference direction, normally the positive x-axis. This angle is often denoted as \( \theta \), while \( r \) represents the radial distance from the origin to the point.

For example, the polar coordinates \((5, \frac{7\pi}{4})\) mean:

• \( r = 5 \): The distance from the origin to the point is 5 units.
• \( \theta = \frac{7\pi}{4} \): The angle formed with the positive x-axis is \(\frac{7\pi}{4}\) radians.

Polar coordinates provide an intuitive way to describe points based on direction and distance, especially in cases involving circular motion or situations where angles are more natural than horizontal or vertical motions.

Rectangular Coordinates

In contrast to polar coordinates, rectangular coordinates (also known as Cartesian coordinates) specify a point in space using a pair of perpendicular axes, usually labeled as the x-axis and y-axis. Each point is determined by the distance from the axes with the values \( x \) and \( y \).

For converting polar coordinates into rectangular coordinates, the following formulas are applied:

• \( x = r \cos(\theta) \)
• \( y = r \sin(\theta) \)

These essential formulas allow us to translate the radial and angular description into a system that uses perpendicular projections, making it easier for certain types of calculations and graphical representations.

When converting the polar coordinates \((5, \frac{7\pi}{4})\) to rectangular coordinates, the point becomes \( \left(\frac{5\sqrt{2}}{2}, -\frac{5\sqrt{2}}{2}\right) \). This successfully translates the angle and radius into a point on a grid-like structure, simplifying the interpretation of the location.

Trigonometric Functions

Trigonometric functions play a key role in the conversion between polar and rectangular coordinates. These functions derive from the relationships within a right triangle and allow us to relate angles to ratios of side lengths.

Key trigonometric functions used in conversions include:

• Cosine (\( \cos \)): Measures the adjacent side over the hypotenuse in a right triangle. It helps find the x-coordinate in polar to rectangular conversion.
• Sine (\( \sin \)): Measures the opposite side over the hypotenuse and is used for calculating the y-coordinate.

These functions utilize the unit circle concept, where angles and lengths can be associated with different quadrants of the circle, allowing conversions between coordinate systems.

For example, in the exercise, \( \cos\left(\frac{7\pi}{4}\right) = \frac{\sqrt{2}}{2} \) is used to find the x-coordinate and \( \sin\left(\frac{7\pi}{4}\right) = -\frac{\sqrt{2}}{2} \) for the y-coordinate. By understanding and applying these trigonometric functions, one can seamlessly convert between polar and rectangular forms.