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Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. In problems involving relative velocity, such as the movement of boat 2, we often use trigonometry to resolve forces or velocities into components aligned with the cardinal directions (i.e., north, east, south, west).

For example, when boat 2 moves at an angle of 30° north of east, we can resolve this velocity into two perpendicular components using trigonometric functions:

• Cosine: This function helps us find the northward component by taking the cosine of the angle (30°) multiplied by the velocity (1.60 m/s), resulting in the formula: \[ v_{n2} = 1.60 \cos(30^{\circ}) \]
• Sine: This function is used to calculate the eastward component by multiplying the sine of the angle (30°) by the velocity, resulting in: \[ v_{e2} = 1.60 \sin(30^{\circ}) \]

Using trigonometry in this way allows us to break down complex motion into simple, one-dimensional motions that are easier to analyze.

The Pythagorean theorem is a fundamental principle in mathematics, helping to find the length of the sides of a right triangle. It expresses a simple relationship between the sides of a right triangle:\[ a^2 + b^2 = c^2 \]where \(a\) and \(b\) are the lengths of the legs, and \(c\) is the hypotenuse.

In the context of the passenger's speed relative to the shore, we use the Pythagorean theorem to determine the resultant velocity from the northward (\(v_{nt}\)) and eastward (\(v_{et}\)) components of velocity. Specifically, after finding the total velocity components:

• Northward component calculated as: \[ v_{nt} = 3.00 + 1.60 \cos(30^{\circ}) \]
• Eastward component combined from boat's and passenger's velocity: \[ v_{et} = 1.60 \sin(30^{\circ}) + 1.20 \]

The Pythagorean theorem allows us to find the total velocity:\[ v_{total} = \sqrt{v_{nt}^2 + v_{et}^2} \]This formula gives us the overall speed of the passenger relative to the shore.

Vectors are quantities that have both magnitude and direction. The velocity of a moving object can be considered a vector, and breaking this vector into components helps to simplify complex movements, especially when analyzing relative velocities.

When we resolve a vector, like the velocity of boat 2, into components:

• The northward component, defined by the cosine of 30°, aligns with the north axis and is expressed by the formula: \[ v_{n2} = 1.60 \cos(30^{\circ}) \]
• The eastward component, determined by the sine of 30°, aligns with the east axis and can be computed as: \[ v_{e2} = 1.60 \sin(30^{\circ}) \]

These components allow us to treat the calculations for each direction separately, simplifying the problem into a more manageable form where basic addition and the application of the Pythagorean theorem fixes the knowledge gaps. The end goal of calculating vector components is to determine the overall effect when they combine.

Velocity addition involves combining velocities to find the resultant velocity of an object, accounting for multiple vectors from different sources. In this problem, we look at the complex motion occurring between boat 1, boat 2, and the walking speed of the passenger.

To find the passenger's overall velocity relative to the shore, follow these steps:

• Add the northward velocity of boat 1 and the northward component of boat 2: \[ v_{nt} = 3.00 + 1.60 \cos(30^{\circ}) \]
• Combine the eastward velocity of the passenger and the eastward component of boat 2: \[ v_{et} = 1.60 \sin(30^{\circ}) + 1.20 \]

This procedure shows how each velocity affects the result, giving us the passenger's overall speed with respect to the shore. By adding these vector components separately, and then combining them using the Pythagorean theorem, we simplify the calculation task.