Navigating the Coordinate Plane: Unraveling Trigonometry in a 3-4-5 Triangle
Imagine a man walking 4 miles east, then 3 miles north, pausing briefly, and continuing his journey by walking 3 miles north and 4 miles east.
How can we determine how far he is from where he started? Let's dive into the details of this fascinating problem and how it involves both the Cartesian plane and trigonometry.
Breaking Down the Journey on the Cartesian Plane
When dealing with such a movement, it's helpful to visualize the journey on a Cartesian plane. The starting point is (0, 0).
First Leg of the Journey
The man walks 4 miles east, which can be represented as moving from (0, 0) to (4, 0).
Second Leg of the Journey After Resting
Then, the man walks 3 miles north, shifting from (4, 0) to (4, 3). This is where he rests for a while.
Third Leg of the Journey
After resting, the man walks 3 miles north and 4 miles east, eventually reaching the point (8, 6).
Calculating the Distance Using the Distance Formula
To find the straight-line distance from the starting point (0, 0) to the final position (8, 6), we can use the distance formula:
Distance √(x? - x?)2 (y? - y?)2
Substituting the coordinates of the starting point (0, 0) and the final position (8, 6), we get:
Distance √(8 - 0)2 (6 - 0)2 √64 36 √100 10 miles
Understanding the Right Triangle
This problem involves a right triangle with sides 3, 4, and 5. The 3-4-5 triangle is a well-known right triangle in trigonometry, and we can use it to solve the problem without complex calculations.
Applying Pythagoras' Theorem
The Pythagorean theorem states that for a right triangle with sides a and b and hypotenuse c, a2 b2 c2. In this case, the sides are 3 and 4, and the hypotenuse is 5.
The distance from the starting point to the final position is therefore 5 miles, which can be calculated as:
Distance √42 32 √16 9 √25 5 miles
Direction of Travel and Trigonometric Angles
Beyond just the distance, we can also determine the direction the man is heading. The triangle formed by the journey has angles that can be calculated using trigonometric functions.
Calculating the Angle
The angle the man is heading can be calculated using the tangent function: tan(θ) opposite/adjacent. In this case, the opposite side is 3 and the adjacent side is 4. The angle is:
θ arctan(3/4) ≈ 36.87°
The direction can be described as a bit east of northeast or 36.87° east of north.
Final Direction and Conclusion
Therefore, the man is 5 miles away from his starting point, and his direction of travel is 36.87° east of north.
This problem not only demonstrates the application of the Pythagorean theorem and trigonometry but also highlights the importance of the Cartesian plane in solving real-world problems related to navigation and distance calculations.