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Trigonometry is a branch of mathematics that explores the relationships between the sides and angles of triangles. Specifically, it is essential in solving problems related to vectors by helping us break down vector magnitudes into different components. When dealing with displacement vectors, trigonometry allows the calculation of both the x and y components using sine and cosine functions.

Why do we use sine and cosine? Imagine placing a right triangle on a coordinate plane, where the hypotenuse represents a vector. The angle between the hypotenuse and the x-axis helps determine how much of the vector extends along the x and y directions.

• The cosine of an angle in a right triangle gives the ratio of the adjacent side (x-component) to the hypotenuse (vector magnitude): \[\cos(\theta) = \frac{x}{r}\]
• Similarly, the sine of the angle gives the ratio of the opposite side (y-component) to the hypotenuse: \[\sin(\theta) = \frac{y}{r}\]

When we focus on an angle like \(30^{\circ}\), trigonometry quickly helps us quantify the projection of the vector along the axes, providing precise components used for further calculations.

A displacement vector represents both the distance and direction from one point to another. It is often visualized as an arrow pointing from an initial location to a final position, embodying the concept of vector magnitude and direction.

In our exercise, the displacement vector \(\vec{r}\) is defined by its length, \(15 \, \text{m}\), and its angle, \(30^{\circ}\). These parameters illustrate how far and at what angle the object has moved from its starting point.

• Magnitude: The length of the vector, here \(15 \, \text{m}\), indicates the extent of the displacement.
• Direction: The angle \(30^{\circ}\) provides the direction of movement relative to the x-axis.

Understanding the full nature of displacement vectors helps us not only solve problems but also offers insights into real-world applications, such as navigation and physics, where direction and distance are crucial.

Vectors are multi-directional and can be split into simpler parts called components. Specifically, the x and y components show how much of the vector acts along the horizontal and vertical axes respectively.

These components are crucial for simplifying calculations in physics and engineering, as they allow vectors to be handled algebraically in equations like forces, velocities, and, as in this case, displacements.

• X-component: Calculated using the cosine function: \[x = r \cdot \cos(\theta)\]With \(r = 15 \text{ m}\) and \(\theta = 30^{\circ}\), the value of \(x\) is approximately \(12.99 \text{ m}\), indicating the horizontal influence of the vector.
• Y-component: Calculated using the sine function: \[y = r \cdot \sin(\theta)\]Using the same values, \(y\) equals \(7.5 \text{ m}\), which tells us the vertical influence.

Breaking down vectors into their x and y components is essential, not only simplifying calculations but also providing a clear picture of how the vector behaves in each direction.