Two radar stations are 50 miles apart. Both are tracking the same plane. At this moment, the angle between station 1 and the plane is 73 degrees and the angle between station 2 and the plane is 46 degrees. How far is the plane from station 2?
By "the angle" I assume you mean the angle between the line from the station to the plane and the line joining the stations. In that case,
Label the diagram as follows.
A = station 1
B = station 2
P = plane
Q = point on AB closes to the plane PQ┴AB)
y = PQ
a = AQ
x = PB (the desired distance)
y/a = tan 73°
y/(50-a) = tan 46°
eliminating the a, we have
y/tan73° = 50 - y/tan46°
y = 39.326
Now, the distance x=PB is found using
y/x = sin 46°
x = 39.326/sin46° = 54.67
To find the distance of the plane from station 2, we can use trigonometry. Let's denote the distance of the plane from station 1 as x, and the distance of the plane from station 2 as y.
First, let's draw a diagram to visualize the situation. We have two radar stations, station 1 and station 2, located 50 miles apart. The plane is at some point in the airspace, and it forms angles of 73 degrees with station 1 and 46 degrees with station 2.
Now let's apply the Law of Sines, which states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides.
By using the Law of Sines, we can write two equations:
sin(73 degrees) / x = sin(46 degrees) / y (equation 1)
sin(180 degrees - 73 degrees - 46 degrees) / 50 = sin(46 degrees) / y (equation 2)
Let's simplify these equations:
sin(73 degrees) / x = sin(46 degrees) / y
x / sin(73 degrees) = y / sin(46 degrees) (equation 3)
sin(61 degrees) / 50 = sin(46 degrees) / y
y = (50 * sin(46 degrees)) / sin(61 degrees) (equation 4)
By using equations 3 and 4, we can solve for y:
x / sin(73 degrees) = (50 * sin(46 degrees)) / sin(61 degrees)
Now, plug in the known angles and solve for x:
x / sin(73 degrees) = (50 * sin(46 degrees)) / sin(61 degrees)
To find the value of x, we need to know the value of sin(73 degrees), sin(46 degrees), and sin(61 degrees). Here's how you can find them:
1. Open a trigonometric calculator or use a scientific calculator that has trigonometric functions.
2. Make sure the calculator is set to degree mode.
3. Enter sin(73) and take the sine of 73 degrees. Write down the result.
4. Repeat the above steps to find sin(46) and sin(61) degrees.
After obtaining the values of sin(73 degrees), sin(46 degrees), and sin(61 degrees), plug them into the equation and solve for x.
x / sin(73 degrees) = (50 * sin(46 degrees)) / sin(61 degrees)
Once you've found the value of x, you can use equation 3 to find the value of y:
y = (x * sin(46 degrees)) / sin(73 degrees)
Now, substitute the value of x in equation 3 to calculate the value of y and find how far the plane is from station 2.