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Answer the following questions about path A in the figure. A city block is a square of l = 125 m on each side, what is the angle of displacement measured east of north? vector goes 3 blocks north and then one block east. Also explain how do you know that it is tangent that we need to calculate and not sin or cos
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Let's solve this vector displacement problem step by step. We have a city with square blocks of 125 meters on each side. Path A goes 3 blocks north and then 1 block east. We need to find the angle of displacement measured east of north. Let me show you the displacement vector on this grid.
Now let's calculate the displacement components. The eastward displacement is 1 block times 125 meters, which equals 125 meters. The northward displacement is 3 blocks times 125 meters, which equals 375 meters. These components form a right triangle where the east component is the opposite side to our angle phi, and the north component is the adjacent side. The angle phi is measured from the north axis toward the east.
Now let's understand why we use the tangent function. In trigonometry, sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, and tangent equals opposite over adjacent. We know the opposite side is 125 meters and the adjacent side is 375 meters. Since we don't need to calculate the hypotenuse, tangent is the most direct function to use. Therefore, tangent of phi equals 125 over 375, which simplifies to 1 over 3.
Now let's calculate the final angle. We have tangent of phi equals 125 over 375, which simplifies to 1 over 3. To find the angle phi, we use the inverse tangent function, also called arctangent. The arctangent of 1 over 3 equals 18.4349 degrees. Rounding to two decimal places, our final answer is 18.43 degrees. This is the angle of displacement measured east of north.
Let's summarize our complete solution. We calculated the displacement components as 125 meters east and 375 meters north. These formed a right triangle where 125 meters is the opposite side and 375 meters is the adjacent side to our angle. Using the tangent function, we found that tangent of phi equals 1 over 3. Taking the arctangent gives us 18.43 degrees. Therefore, the angle of displacement measured east of north is 18.43 degrees. This demonstrates why tangent is the most direct trigonometric function when we know both perpendicular sides of a right triangle.