Problem 32 - Trigonometry

Question

Distance \(A\) boy is riding his motorcycle on a road that runs east and west. He leaves the road at a service station and rides \(5.25\) miles in the direction \(\mathrm{N} \mathrm{} 15.5^{\circ} \mathrm{E}\). Then he turns to his right and rides \(6.50\) miles back to the road, where his motorcycle breaks down. How far will he have to walk to get back to the service station?

Step-by-Step Solution

Answer

The boy needs to walk approximately 6.64 miles back to the service station.

Step 1: Understand the Problem

The boy rides from the service station in a direction that makes an angle of \(15.5^\circ\) north of east. He rides \(5.25\) miles in this direction, then turns right and rides \(6.50\) miles until he reaches the road again. We need to find the straight distance back to the service station.

Step 2: Visualize with a Diagram

Imagine a right triangle where the service station is the origin. The boy rides \(5.25\) miles at \(15.5^\circ\) north of east (hypotenuse), then turns \(90^\circ\) to the right and rides \(6.50\) miles to rejoin the road (adjacent side). We need to find the length of the third side of this triangle (opposite side), which represents the distance back to the service station.

Step 3: Calculate East-West Component

The east-west component of the boy's initial journey can be calculated using the cosine of \(15.5^\circ\). \[x = 5.25 \cdot \cos(15.5^\circ)\]Calculate this value to find the eastward displacement.

Step 4: Calculate North-South Component

The north-south component of the initial ride can be found using the sine of \(15.5^\circ\). \[y = 5.25 \cdot \sin(15.5^\circ)\]Calculate this value to find the northward displacement.

Step 5: Verify Triangle Side Orientation

The second leg of \(6.50\) miles is completely along the road, it means it's along the east-west line. Its orientation and the initial northward component intersect perpendicularly, forming a right triangle.

Step 6: Use Pythagorean Theorem to Find Distance

Now, calculate the direct distance (opposite side of the triangle) from the terminal point back to the service station using the Pythagorean theorem.\[z = \sqrt{(5.25 \cdot \cos(15.5^\circ) + 6.50)^2 + (5.25 \cdot \sin(15.5^\circ))^2}\]Solve for \(z\) to determine the walking distance.

Key Concepts

Right Triangle

When dealing with a right triangle, it is important to understand its unique properties. A right triangle is a type of triangle that has one angle exactly equal to 90 degrees. This creates a perfect perpendicular intersection with its sides.

The sides of a right triangle include:

• Hypotenuse: This is the longest side and is opposite the right angle.
• Adjacent Side: This side is next to the angle of interest, but not the hypotenuse.
• Opposite Side: This is the side opposite the angle of interest.

In our case, when the boy rides his motorcycle, he travels along a path that forms a right triangle with his starting point. By converting his motion into triangle components, we are able to calculate the distance directly back to the service station.

Pythagorean Theorem

The Pythagorean theorem is a fundamental formula used to find the length of one side in a right triangle when the other two sides are known. It is defined as:

\[ c^2 = a^2 + b^2 \]

Where:

• \(c\) is the length of the hypotenuse.
• \(a\) and \(b\) are the lengths of the other two sides.

In practical terms, you can think of the Pythagorean theorem as a tool that connects the dots back and forth across a triangle, allowing you to "see" the unseen side lengths.

For the task at hand, we use this theorem to calculate the direct distance that the boy needs to walk back to the service station. By substituting the east-east and north-south distances into the formula, we solve for the required direct walking path.

Sine and Cosine Functions

Sine and cosine are trigonometric functions used to relate the angles and sides of a right triangle. Knowing just one angle besides the 90-degree corner allows you to use these functions successfully.

• Cosine Function: For a given angle \(\theta\), the cosine function can help find the length of the adjacent side of a right triangle using the formula \(\cos(\theta) = \frac{adjacent}{hypotenuse}\).
• Sine Function: Similarly, the sine function helps determine the length of the opposite side, defined as \(\sin(\theta) = \frac{opposite}{hypotenuse}\).

In this exercise, when the boy rides north of east, we employ these functions to calculate both the north-south and east-west components of his journey. By doing so, we divide his route into simple vector components that facilitate solving for the unknown distance back to the point of origin.