Problem 18 - Calculus Early Transcendentals

Question

Express the following polar coordinates in Cartesian coordinates. \(\left(2, \frac{7 \pi}{4}\right)\)

Step-by-Step Solution

Answer

Question: Convert the polar coordinates (2, 7π/4) to Cartesian coordinates. Answer: The Cartesian coordinates are (√2, -√2).

Step 1: Find \(\cos(\theta)\) and \(\sin(\theta)\)

We want to find \(\cos\left(\frac{7\pi}{4}\right)\) and \(\sin\left(\frac{7\pi}{4}\right)\). Since \(\frac{7\pi}{4}\) is in the fourth quadrant, we know that \(\cos(\theta) > 0\) and \(\sin(\theta) < 0\). Let's use the reference angle \(\alpha = \frac{pi}{4}\) and find the cosine and sine values for the given angle. We know that \(\cos\left(\frac{\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}\), so: \(\cos\left(\frac{7\pi}{4}\right) = \frac{1}{\sqrt{2}}\) \(\sin\left(\frac{7\pi}{4}\right) = -\frac{1}{\sqrt{2}}\)

Step 2: Convert polar coordinates to Cartesian coordinates

Now we can use the formulas for polar to Cartesian conversion and plug in the values we found in step 1: \(x = r \cdot \cos(\theta) = 2 \cdot \frac{1}{\sqrt{2}} = \sqrt{2}\) \(y = r \cdot \sin(\theta) = 2 \cdot -\frac{1}{\sqrt{2}} = -\sqrt{2}\) So the Cartesian coordinates are \((\sqrt{2}, -\sqrt{2})\).

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. They are essential in converting polar coordinates, which consist of a radius and an angle, to Cartesian coordinates that use x and y coordinates.

The two primary trigonometric functions used in this conversion are cosine (\( \cos \)) and sine (\( \sin \)). These functions are derived from right triangles and relate directly to circular motion.

• Cosine (\( \cos \)): This function gives the x-coordinate (horizontal value) when using the unit circle, where the circle's radius is 1. It represents the adjacent side of a right triangle over the hypotenuse.
• Sine (\( \sin \)): This function provides the y-coordinate (vertical value) on the unit circle. It is represented by the opposite side over the hypotenuse in a right triangle.

In the conversion from polar to Cartesian coordinates, these functions determine how much of the radius is applied in each direction (x and y), making them crucial in calculating exact positions on a plane.

Reference Angles

A reference angle is the smallest angle that a terminal side of an angle makes with the x-axis. In trigonometry, the reference angle aids in determining the values of the trigonometric functions of angles greater than 90 degrees. It can be particularly useful when the angle is located in different quadrants.

For instance, the polar angle \(\frac{7\pi}{4}\) radians is found in the fourth quadrant. To calculate, we determine its reference angle, \(\alpha\), which here is \(\frac{\pi}{4}\) radians. This is achieved by subtracting the angle from \(2\pi\), since the whole unit circle is \(2\pi\) radians.

• The reference angle ensures that the trigonometric functions for \(\frac{7\pi}{4}\) and \(\frac{\pi}{4}\) have the same absolute values, though signs might differ depending on the quadrant.
• Using understanding from right triangles, these reference angles simplify the calculation of sine and cosine values.

By understanding reference angles, solving problems with higher angles becomes much more manageable, ensuring that all computations remain within the familiar 0 to \(\frac{\pi}{2}\) range.

Quadrants in Trigonometry

The unit circle is divided into four quadrants, each containing unique characteristics that influence the sign of trigonometric functions. This system helps to understand angles and their trigonometric values better.

The quadrants are labeled in an anti-clockwise direction starting from the positive x-axis:

• First Quadrant: Angles from 0 to \(\frac{\pi}{2}\). Both sine and cosine values are positive.
• Second Quadrant: Angles from \(\frac{\pi}{2}\) to \(\pi\). Here, sine remains positive, but cosine is negative.
• Third Quadrant: Angles from \(\pi\) to \(\frac{3\pi}{2}\). Both sine and cosine are negative.
• Fourth Quadrant: Angles from \(\frac{3\pi}{2}\) to \(2\pi\). Sine is negative, while cosine is positive.

For \(\frac{7\pi}{4}\), as previously stated, it lies in the fourth quadrant. Therefore, we know its cosine is positive, as seen with \(\cos\left(\frac{7\pi}{4}\right) = \frac{1}{\sqrt{2}}\), while sine is negative, as \(\sin\left(\frac{7\pi}{4}\right) = -\frac{1}{\sqrt{2}}\). Understanding these quadrant rules is essential for solving trigonometric problems effectively.