A lighthouse is east of a Coast Guard patrol boat. The Coast Guard station is 20 km north of the lighthouse. The radar officer aboard the boat measures the angle between the lighthouse and the station to be 23°. How far is the boat from the station?
draw a diagram to see that
20/d = sin23°
To find the distance between the boat and the station, we can use trigonometry.
Let's denote the distance between the boat and the station as "x".
We have a right triangle formed by the boat, the station, and the lighthouse. The angle between the boat and the lighthouse (let's call it angle A) is given as 23°.
In a right triangle, 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. So, in this case, we have:
sin(A) = opposite / hypotenuse
In our triangle, angle A is the angle between the boat and the lighthouse, the distance between the boat and the station is the opposite side, and the distance between the boat and the lighthouse is the hypotenuse.
Therefore, we have:
sin(23°) = x / 20 km
To find "x", we can rearrange the equation as follows:
x = sin(23°) * 20 km
Now, let's calculate the value of "x":
x ≈ sin(23°) * 20 km
x ≈ 0.3907 * 20 km
x ≈ 7.814 km
Thus, the boat is approximately 7.814 km away from the station.
To find the distance between the boat and the station, we can use trigonometry. We have a right triangle formed by the boat, the lighthouse, and the station.
Let's assume that the distance between the boat and the station is represented by the variable 'x'. We can use the tangent function to solve the problem.
First, we need to find the length of the opposite side of the right triangle, which is the distance between the boat and the lighthouse. Since the angle measures 23° and the side opposite the angle is 20 km, we can use the trigonometric function tangent:
tan(23°) = opposite / adjacent
tan(23°) = x / 20
To find x, we can rearrange the equation:
x = 20 * tan(23°)
Using a calculator, we can find the value of tan(23°) ≈ 0.4245:
x = 20 * 0.4245
x ≈ 8.49 km
Therefore, the boat is approximately 8.49 km away from the station.