Introduction
When analyzing the flight path of an aircraft, understanding the components of its journey is crucial for navigation and analysis purposes. This article will guide you through a detailed process using trigonometry to calculate how far a plane is from its starting point after a series of flights, and in which direction it is located.
Understanding Trigonometry in Flight Path Analysis
Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. In the context of a plane's flight path, we can break down its journey into individual segments, calculate the horizontal and vertical components of each segment, and then combine these components to determine the overall displacement from the starting point.
Breaking Down the First Leg of the Flight
The first leg of the flight involves a plane traveling 170 km at an angle of 68 degrees east of north.
Distance: 170 km Angle: 68 degrees east of northWe need to convert this into Cartesian coordinates (x, y), where north corresponds to the positive y-axis and east corresponds to the positive x-axis.
To find the x and y components, we use the trigonometric functions sine and cosine:
[x_1 170 cdot sin(68^circ)]
[y_1 170 cdot cos(68^circ)]
Calculating these values, we get:
[x_1 170 cdot sin(68^circ) approx 170 cdot 0.9272 approx 157.63 , text{km east}]
[y_1 170 cdot cos(68^circ) approx 170 cdot 0.3746 approx 63.66 , text{km north}]
Breaking Down the Second Leg of the Flight
The second leg of the flight involves a plane traveling 230 km at an angle of 48 degrees south of east.
Distance: 230 km Angle: 48 degrees south of eastFor this leg, the components are calculated as:
[x_2 230 cdot cos(48^circ)]
[y_2 -230 cdot sin(48^circ) , text{(negative because it is south)}
Calculating these values, we get:
[x_2 230 cdot cos(48^circ) approx 230 cdot 0.6691 approx 153.91 , text{km east}]
[y_2 -230 cdot sin(48^circ) approx -230 cdot 0.7431 approx -170.90 , text{km south}]
Combining the Components
Now, we can add the x and y components of both legs to find the total displacement:
[x_{text{total}} x_1 x_2 157.63 153.91 approx 311.54 , text{km}]
[y_{text{total}} y_1 y_2 63.66 - 170.90 approx -107.24 , text{km}]
The total distance from the airport can be calculated using the Pythagorean theorem:
[d sqrt{x_{text{total}}^2 y_{text{total}}^2} sqrt{311.54^2 (-107.24)^2} approx sqrt{96861.24 11498.78} approx sqrt{108360.02} approx 329.00 , text{km}]
Calculating the Direction
To find the direction, we use the arctangent function to calculate the angle relative to the east:
[theta tan^{-1}left(frac{y_{text{total}}}{x_{text{total}}}right) tan^{-1}left(frac{-107.24}{311.54}right)]
[theta approx tan^{-1}(-0.344) approx -18.92^circ]
This angle is measured clockwise from the positive x-axis (east), indicating that the direction is 18.92 degrees south of east.
Final Result
After the entire flight, the plane will be approximately 329 km from the airport, in a direction approximately 18.92 degrees south of east.
Conclusion
Moving through a series of calculated steps in trigonometry demonstrates how we can precisely determine a plane's final position from its starting point. Clear understanding and application of these techniques are essential for both practical navigation and theoretical analysis.