Assume two radar stations that are 25 miles apart are both tracking a plane. At a given moment, the angle between Station 1 and the plane is 73 °, while the angle between Station 2 and the plane is 46 °. How far is the plane from Station 2?
is the plane between the two stations, or on the same side of both?
Draw a diagram and consider using the tan or cot function.
To find the distance of the plane from Station 2, we can use trigonometry. Since we have the angles and the distance between the two radar stations, we can apply the sine function.
1. Draw a diagram: Start by drawing a diagram of the situation. Label the two radar stations as Station 1 and Station 2, and label the distance between them as 25 miles. Also, mark the plane.
2. Identify the known information: From the question, we know that the angle between Station 1 and the plane is 73 °, and the angle between Station 2 and the plane is 46 °. Additionally, we know the distance between the two radar stations is 25 miles.
3. Set up the equation: Using the sine function, we know that the sine of an angle is equal to the length of the opposite side divided by the length of the hypotenuse. In this case, the opposite side is the distance from Station 2 to the plane, and the hypotenuse is the distance between the two radar stations.
sin(46 °) = (distance from Station 2 to the plane) / 25 miles
4. Solve the equation: Rearrange the equation to solve for the distance from Station 2 to the plane:
(distance from Station 2 to the plane) = 25 miles × sin(46 °)
Using a calculator, evaluate sin(46 °), which is approximately 0.7193:
(distance from Station 2 to the plane) ≈ 25 miles × 0.7193 ≈ 17.9825 miles
5. Answer: The plane is approximately 17.9825 miles away from Station 2.