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When solving problems in physics, especially in air navigation, it's essential to break down vectors into their respective components. This method, known as vector decomposition, allows us to better understand and manage the different forces and directions involved. In our problem, we decompose the velocity of the airplane into north and west components because the plane is navigating in a windy environment.
For instance, if the plane needs to head 22 degrees west of true north, we need to calculate how much of its velocity points north and how much points west. This helps in understanding how the wind affects the plane's movement and is vital for accurately determining wind velocity.

Decomposing the velocity of an airplane into its north and west components allows us to see how these directions interact. In the given problem, the plane has a known velocity in still air. We use trigonometric functions, notably sine and cosine, to find these components.
The formulas we use here are:

• Northward velocity component: \(v_{\text{plane, north}} = v_{\text{plane}} \times \text{cos}(22^\text{°})\)
• Westward velocity component: \(v_{\text{plane, west}} = v_{\text{plane}} \times \text{sin}(22^\text{°})\)

By using these equations, we calculate how much of the plane's speed aims north and west. This breakdown provides the precise data needed to figure out the wind's impact.

Understanding the Pythagorean Theorem is crucial when dealing with wind velocities. In our context, it helps to find the magnitude of the wind's velocity when we know its north and west components. The Pythagorean Theorem states that in a right-angled triangle, the square of the hypotenuse is the sum of the squares of the other two sides.
Mathematically, it is represented as:
\(
v_{\text{wind}} = \text{sqrt}(v_{\text{wind, north}}^2 + v_{\text{wind, west}}^2)
\) .
Using this theorem, after determining the north and west components of the wind's velocity, we calculate the overall wind speed. This provides us with the wind's magnitude, essential for accurate air navigation.

Trigonometric functions play a fundamental role in solving physics problems involving angles and directions. In the exercise, we used these functions to find the components of the airplane's velocity. Utilizing sine and cosine, we can accurately split vectors into their respective parts.
Here's an overview of their roles:

• **Cosine function**: Used for finding the adjacent side of a right-angled triangle, which, in this case, is the north component of the velocity.
• **Sine function**: Used for finding the opposite side, which is the west component of the velocity.

We start with the airplane's known velocity in still air and multiply it by the cosine of 22 degrees to find the north component and by the sine of 22 degrees to get the west component. Trigonometry thus enables precise and practical solutions to real-world navigation problems.