Problem 30 - Precalculus

Question

Convert each point to exact rectangular coordinates. $$\left(5,315^{\circ}\right)$$

Step-by-Step Solution

Answer

\( \left(\frac{5\sqrt{2}}{2}, -\frac{5\sqrt{2}}{2}\right) \)

Step 1: Understand the Polar Coordinates

We are given a point in polar coordinates, which is represented as \((r, \theta)\), where \(r\) is the distance from the origin and \(\theta\) is the angle from the positive x-axis. The given point is \((5, 315^{\circ})\).

Step 2: Convert Angle to Radians

Before conversion, it is useful to think of the angle in radians. However, in practical terms of this exercise, since 315° will be directly used as a calculation for sine and cosine without converting to radians. We will use these measures directly as they are universally understood in both radians and degrees for trigonometric functions.

Step 3: Calculate the x-coordinate

The x-coordinate in rectangular coordinates is found using the formula: \[ x = r \cdot \cos(\theta) \]Substitute the given values:\[ x = 5 \cdot \cos(315^{\circ}) \] Since \(\cos(315^{\circ}) = \cos(360^{\circ} - 45^{\circ}) = \cos(45^{\circ}) = \frac{\sqrt{2}}{2} \), we have:\[ x = 5 \cdot \frac{\sqrt{2}}{2} = \frac{5\sqrt{2}}{2} \].

Step 4: Calculate the y-coordinate

The y-coordinate is calculated using the formula:\[ y = r \cdot \sin(\theta) \]Substitute the given values:\[ y = 5 \cdot \sin(315^{\circ}) \]Since \(\sin(315^{\circ}) = \sin(360^{\circ} - 45^{\circ}) = -\sin(45^{\circ}) = -\frac{\sqrt{2}}{2} \),we have:\[ y = 5 \cdot \left(-\frac{\sqrt{2}}{2}\right) = -\frac{5\sqrt{2}}{2} \].

Step 5: Write Rectangular Coordinates

Combine the calculated x-coordinate and y-coordinate to express the point in rectangular coordinates. The point \((5, 315^{\circ})\) in rectangular coordinates is:\[ \left(\frac{5\sqrt{2}}{2}, -\frac{5\sqrt{2}}{2}\right) \].

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, represent points in the plane using a pair \(x, y\), which corresponds to the horizontal and vertical distances from the origin, respectively. This system is named after the mathematician René Descartes, and it forms the basis of many mathematical applications in geometry and calculus. In this coordinate system:

• \(x\) is the distance along the horizontal x-axis, with positive values to the right and negative values to the left from the origin.
• \(y\) is the distance along the vertical y-axis, with positive values upwards and negative values downwards from the origin.

To convert from polar to rectangular coordinates, we use the relationships between the polar coordinates \(r, \theta\) and rectangular coordinates:

• The x-coordinate is found by \(x = r \cdot \cos(\theta)\).
• The y-coordinate is found by \(y = r \cdot \sin(\theta)\).

These formulas allow us to express points initially given in polar coordinates in the familiar (x, y) format we commonly use in everyday settings such as maps or grids.

Trigonometric Functions

Trigonometric functions relate the angles and sides of right-angled triangles, and they are fundamental in converting between different coordinate systems, such as polar and rectangular coordinates. The primary functions are:

• Cosine (cos), which represents the adjacent side over the hypotenuse in a right triangle.
• Sine (sin), which represents the opposite side over the hypotenuse.

In our conversion from polar to rectangular coordinates:

• The cosine function is used to determine the x-coordinate, which accounts for the horizontal component of the point's position.
• The sine function is used to find the y-coordinate, representing the vertical component of the point's position.

For example, when dealing with an angle like \(315^\circ\), equivalent trigonometric values are used because the unit circle properties allow us to simplify these calculations. \(\cos(315^\circ) = \frac{\sqrt{2}}{2}\) and \(\sin(315^\circ) = -\frac{\sqrt{2}}{2}\) due to the symmetry and periodicity of the trigonometric functions, relating this angle to the well-known angle of \(45^\circ\).

Degree to Radian Conversion

The conversion between degrees and radians is crucial in mathematics, especially when performing calculations involving angles. Degrees and radians are two units used to measure angles. Here's a little breakdown:

• Degrees measure the angle based on a full circle being \(360^\circ\).
• Radians measure the angle based on a full circle being \(2\pi\) radians.

To convert an angle from degrees to radians:

• Use the formula: \(\theta \, (\text{in radians}) = \theta \times \frac{\pi}{180}\).

In the case of our exercise, though it wasn't necessary to convert, understanding radians is helpful for further advanced applications in mathematics and physics, where radian measure is preferred due to its natural association with the unit circle and trigonometric calculations.