Suppose A Lookout Station { A $}$ Is 50 Miles West Of Ember Island. An Earth Kingdom Battleship { B $}$ Is Due North Of Ember Island. Lookout Station { A $}$ Sees The Ship { B $}$ At A Bearing Of [$ N

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Solving a Trigonometry Problem: Finding the Distance Between Two Points

In this article, we will explore a problem involving trigonometry and geometry. We will use the concept of bearings and angles to find the distance between two points on a coordinate plane. The problem is as follows: Suppose a lookout station { A $}$ is 50 miles west of Ember Island. An Earth Kingdom battleship { B $}$ is due north of Ember Island. Lookout station { A $}$ sees the ship { B $}$ at a bearing of { N 60^\circ W $}$. We need to find the distance between the lookout station and the battleship.

To solve this problem, we need to understand the concept of bearings and angles. A bearing is an angle measured clockwise from north. In this case, the bearing of the battleship from the lookout station is { N 60^\circ W $}$, which means that the ship is 60 degrees west of north. We can represent this angle as a point on a coordinate plane, with the x-axis representing the east-west direction and the y-axis representing the north-south direction.

Let's draw a diagram to visualize the problem. We can represent the lookout station as point { A $}$ at (50, 0) and the battleship as point { B $}$ at (0, y). Since the bearing of the battleship from the lookout station is { N 60^\circ W $}$, we can draw a line from point { A $}$ to point { B $}$ at an angle of 60 degrees.

To find the distance between the lookout station and the battleship, we can use the concept of the sine and cosine functions. We can draw a right triangle with the line from point { A $}$ to point { B $}$ as the hypotenuse, and the x-axis and y-axis as the other two sides. The angle between the x-axis and the line from point { A $}$ to point { B $}$ is 60 degrees.

We can use the sine and cosine functions to find the distance between the lookout station and the battleship. The sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. In this case, the sine of 60 degrees is equal to the ratio of the length of the side opposite the angle (the y-coordinate of point { B $}$) to the length of the hypotenuse (the distance between point { A $}$ and point { B $}$).

We can calculate the distance between the lookout station and the battleship using the sine and cosine functions. We know that the sine of 60 degrees is equal to { \frac{\sqrt{3}}{2} $}$. We can set up an equation using the sine function:

sin⁡(60∘)=y502+y2\sin(60^\circ) = \frac{y}{\sqrt{50^2 + y^2}}

We can solve for y by multiplying both sides of the equation by { \sqrt{50^2 + y^2} $}$:

32502+y2=y\frac{\sqrt{3}}{2} \sqrt{50^2 + y^2} = y

We can square both sides of the equation to eliminate the square root:

34(502+y2)=y2\frac{3}{4} (50^2 + y^2) = y^2

We can simplify the equation by multiplying both sides by 4:

3(502+y2)=4y23(50^2 + y^2) = 4y^2

We can expand the left side of the equation:

30000+3y2=4y230000 + 3y^2 = 4y^2

We can subtract 3y^2 from both sides of the equation:

30000=y230000 = y^2

We can take the square root of both sides of the equation to solve for y:

y=30000y = \sqrt{30000}

We can simplify the right side of the equation:

y=173.21y = 173.21

Now that we have found the y-coordinate of point { B $}$, we can find the distance between the lookout station and the battleship using the Pythagorean theorem. The Pythagorean theorem states that the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the other two sides.

We can calculate the distance between the lookout station and the battleship using the Pythagorean theorem:

d=502+y2d = \sqrt{50^2 + y^2}

We can substitute the value of y that we found earlier:

d=502+173.212d = \sqrt{50^2 + 173.21^2}

We can simplify the right side of the equation:

d=2500+30000d = \sqrt{2500 + 30000}

We can simplify the right side of the equation:

d=35000d = \sqrt{35000}

We can simplify the right side of the equation:

d=185.84d = 185.84

In this article, we used the concept of bearings and angles to find the distance between two points on a coordinate plane. We drew a diagram to visualize the problem and used the sine and cosine functions to find the distance between the lookout station and the battleship. We calculated the distance using the Pythagorean theorem and found that the distance between the lookout station and the battleship is approximately 185.84 miles.
Q&A: Solving a Trigonometry Problem

In our previous article, we explored a problem involving trigonometry and geometry. We used the concept of bearings and angles to find the distance between two points on a coordinate plane. In this article, we will answer some common questions that readers may have about the problem.

A: The bearing of the battleship from the lookout station is { N 60^\circ W $}$, which means that the ship is 60 degrees west of north.

A: We can represent the lookout station as point { A $}$ at (50, 0) and the battleship as point { B $}$ at (0, y).

A: The angle between the x-axis and the line from point { A $}$ to point { B $}$ is 60 degrees.

A: We can use the sine and cosine functions to find the distance between the lookout station and the battleship. The sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. In this case, the sine of 60 degrees is equal to the ratio of the length of the side opposite the angle (the y-coordinate of point { B $}$) to the length of the hypotenuse (the distance between point { A $}$ and point { B $}$).

A: We can calculate the distance between the lookout station and the battleship using the Pythagorean theorem. The Pythagorean theorem states that the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the other two sides.

A: The distance between the lookout station and the battleship is approximately 185.84 miles.

A: The bearing of the battleship from the lookout station is significant because it allows us to determine the direction of the ship from the lookout station. In this case, the bearing is { N 60^\circ W $}$, which means that the ship is 60 degrees west of north.

A: We can use the concept of bearings and angles to solve problems involving trigonometry and geometry by drawing diagrams to visualize the problem and using the sine and cosine functions to find the distance between two points on a coordinate plane.

In this article, we answered some common questions that readers may have about the problem of finding the distance between two points on a coordinate plane using the concept of bearings and angles. We hope that this article has been helpful in clarifying any confusion and providing a better understanding of the problem.