Problem 68 - Precalculus : Building Concepts and Connections

Question

A boat departs its starting point and travels 4 miles north and 3 miles west. Determine its current location in polar coordinates, using the starting point as the origin. Use a scientific calculator to approximate \(\theta,\) in radians, to three decimal places.

Step-by-Step Solution

Answer

The location of the boat in polar coordinates is \((5,2.214)\).

Step 1: Compute Distance

We will first find r, the distance from the starting point. We can do this by summing the squares of x and y distances and taking the square root. This is represented by the following equation: \(r = \sqrt{x^2+y^2} = \sqrt{(-3)^2 + 4^2} = \sqrt{9+16} = \sqrt{25}= 5\) So, the distance from the starting point is 5 miles.

Step 2: Compute Angle

Next, let's find the angle \(\Theta\). We will use the formula: \(\Theta = \arctan(\frac{y}{x})\). However, since the x-coordinate is negative (we're going west), our angle will be in the second quadrant. So, the correct formula will be \(\Theta = \pi + \arctan(\frac{y}{x})\). Now we substitute y = 4 (since we're going north) and x = -3 (since we're going west): \(\Theta = \pi + \arctan(\frac{4}{-3})\). This simplification depends on a technological tool like a calculator which will give \(\Theta = 2.214\) (in radians) when rounded to three decimal places.

Step 3: Construct Polar Coordinate

Now, we can construct the polar coordinate using our results from the previous two steps. Polar coordinates are in the form \((r, \Theta)\) where r is the distance and \(\Theta\) is the angle. Hence, the answer is \((5,2.214)\).

Key Concepts

Distance Calculation

In polar coordinates, determining the distance is the first step in locating a position. This involves using a method known as the distance calculation. Here, we use the Pythagorean theorem to find this distance. The theorem relates to a right-angled triangle, and it allows us to calculate the length of the hypotenuse if the other two sides are known.
The position change is represented as moving 4 miles north and 3 miles west. This movement forms a right-angled triangle where:

• The northward movement represents one leg of the triangle (4 miles).
• The westward movement represents another leg (-3 miles as we consider west being a negative direction).

The hypotenuse, which is our distance from the origin or starting point, is calculated as:\[r = \sqrt{(-3)^2 + 4^2} = \sqrt{9+16} = \sqrt{25} = 5\]Thus, the distance the boat has traveled in the polar coordinate system is 5 miles.

Trigonometry

Trigonometry becomes super useful when determining angles in polar coordinates. Specifically, it helps when we're dealing with right-angled triangles, like in our boat scenario.
The primary goal here is finding the angle as the boat has traveled a certain distance in both the north and west direction. The angle of interest is usually found using the inverse tangent function, often written as \(\arctan\). It helps in finding the angle \(\Theta\) where the height or the opposite side is known, along with the base or the adjacent side:

• The height corresponds to the northward travel (4 miles).
• The base relates to westward travel (-3 miles, since we're moving west).

Since the boat is going west (-3 miles), this angle lies in the second quadrant. We adjust for this by recalculating \(\Theta\) with:\[\Theta = \pi + \arctan\left(\frac{4}{-3}\right)\]Trigonometric functions allow us to compute accurate angle measurements, which are essential in converting to polar coordinates.

Angle Measurement

Angle measurement in polar coordinates involves determining the angle \(\Theta\), which indicates the direction of the point from the origin. This often involves conversion to radians—a measurement commonly used in mathematics because it relates angles to the circumference of a circle.
In the boat's example, we need an angle in the second quadrant since westward and northward travel imply this position.

• Use the inverse tangent function to calculate an initial angle from y-x values.
• Adjust for the quadrant by adding \(\pi\), crucial for your angle to reflect the correct direction.
• Radian measure facilitates these calculations, especially in calculators, often providing more precision than degrees.

For our exercise, the final computed angle was approximately \(2.214\) radians, marking the direction from the starting point to the current location on the circle's grid. This precise angle measurement, combined with distance, yields a complete polar coordinate location, \((5, 2.214)\), for the boat's position.