Problem 29 - Precalculus

Question

Convert each point to exact rectangular coordinates. $$\left(-1,135^{\circ}\right)$$

Step-by-Step Solution

Answer

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

Step 1: Convert angle to radians

To convert the given angle from degrees to radians, we use the conversion factor: \( \frac{\pi}{180^\circ} \). Thus, \( 135^\circ = 135 \times \frac{\pi}{180} = \frac{3\pi}{4} \) radians.

Step 2: Use polar to rectangular conversion formulas

The rectangular coordinates \((x, y)\) can be found using the formulas: \(x = r \cos(\theta) \) and \( y = r \sin(\theta) \), where \( r = -1 \) and \( \theta = \frac{3\pi}{4} \).

Step 3: Calculate x-coordinate

Substitute \( r = -1 \) and \( \theta = \frac{3\pi}{4} \) into the \( x \) formula: \[ x = -1 \cdot \cos\left(\frac{3\pi}{4}\right) = -1 \cdot \left(-\frac{\sqrt{2}}{2}\right) = \frac{\sqrt{2}}{2}. \]

Step 4: Calculate y-coordinate

Substitute \( r = -1 \) and \( \theta = \frac{3\pi}{4} \) into the \( y \) formula: \[ y = -1 \cdot \sin\left(\frac{3\pi}{4}\right) = -1 \cdot \frac{\sqrt{2}}{2} = -\frac{\sqrt{2}}{2}. \]

Step 5: Final rectangular coordinates

Now we have the rectangular coordinates as \( \left(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right) \). Therefore, the point \( (-1, 135^\circ ) \) converts to \( \left(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right) \) in rectangular coordinates.

Key Concepts

Radians and Degrees

Understanding the relationship between radians and degrees is crucial when dealing with angles in different mathematical contexts. Degrees are often used in everyday settings, like describing temperatures or angles in simple geometry. However, in more advanced fields such as calculus or trigonometry, radians become the preferred measurement. This is mainly because radians provide a natural measure of angle, related directly to the radius of a circle.

To convert degrees to radians, we use the formula:

• Multiply the degree value with the conversion factor \( \frac{\pi}{180} \).

For example, when converting 135 degrees to radians, the calculation would be:

• \( 135 \times \frac{\pi}{180} = \frac{3\pi}{4} \) radians.

By mastering this simple conversion technique, you will be better equipped to switch between these systems effortlessly. Switching is often necessary when performing calculations in a range of mathematical contexts.

Trigonometric Formulas

Trigonometric formulas are essential for transforming points from one coordinate system to another, like from polar to rectangular coordinates. The two basic formulas you need to remember for these transformations are:

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

Here, \( r \) represents the distance from the origin, and \( \theta \) is the angle in radians.

For converting the point (-1, \(135^\circ\)) to rectangular coordinates, we first convert the angle to radians (\( \frac{3\pi}{4} \)). Then apply the formulas:

• \( x = -1 \cdot \cos\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2} \)
• \( y = -1 \cdot \sin\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} \)

Have these formulas at your fingertips whenever you're tackling problems involving trigonometry or coordinate conversions.

Coordinate Systems

Understanding different coordinate systems is like learning a new language for describing positions. In mathematics, two commonly used systems are polar and rectangular (or Cartesian) coordinates.

- **Polar coordinates** use a radius and an angle to define a point. The radius indicates how far the point is from the origin, and the angle shows the direction relative to the positive x-axis.- **Rectangular coordinates** describe a point using horizontal and vertical displacements (\(x, y\)).
When converting from polar to rectangular coordinates, it's important to understand the underlying geometry. Imagine a right triangle formed by dropping a perpendicular from the point to the x-axis. Here, \(r\) is the hypotenuse, \(x\) is adjacent to \(\theta\), and \(y\) is opposite to \(\theta\). By understanding these relationships, you can visualize how the trigonometric formulas actually work to translate one system into the other, making this mathematical transition more intuitive.