Refer to the diagram below
Point A = plane's starting point at the origin (0,0)
Point B = location in which the plane changes course
Point C = final destination of the plane
The goal is to find the location of point C which tells us directly how far east and north we have traveled
The x coordinate of point C tells us how far east the plane has traveled
The y coordinate of point C tells us how far north the plane has traveled
Length of segment AB = 110 miles
Length of segment BC = 150 miles
These two lengths will be used as r values (in the two sections below).
Points D, E, F, G are helper points to set up the angles below
Angle DAB = 17 degrees (from N 17 E meaning "face north, then turn 17 degrees to the east")
Angle EAB = 73 degrees (90 - angleDAB = 90 - 17 = 73)
Angle FBC = 48 degrees (from N 48 E meaning "face north, then turn 48 degrees to the east")
Angle GBC = 42 degrees (90 - angleFBC = 90 - 48 = 42)
The coordinates of points D, E, F, G are not important.
We will use these two angles
Angle EAB = 73 degrees
Angle GBC = 42 degrees
when it comes to figuring out the locations of point B and point C. Specifically these are the theta values used.
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Let's start off figuring out point B
Point A is at (0,0). Using trig, we can apply these polar form equations
x = r*cos(theta)
y = r*sin(theta)
to determine the point (x,y) where B will be located. The 'r' is the radius or the distance traveled while theta is the angle in which we are pointed. That angle is 73 degrees (it's the angle made with the horizontal axis as the diagram shows) and r = 110 is the distance we travel going from A to B. Therefore,
x = r*cos(theta) = 110*cos(73) = 32.16089 approximately
And,
y = r*sin(theta) = 110*sin(73) = 105.193523 approximately
Point B is located at the approximate location of (x,y) = (32.16089, 105.193523)
So far the plane has traveled roughly 105 miles north and 32 miles east
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Let's figure out the location of point C
Again we will use these formulas
x = r*cos(theta)
y = r*sin(theta)
however we will need to add on the x and y coordinates from point B, so we get the proper offset.
r = 150 miles traveled from B to C
theta = 42 degrees (angle GBC in the diagram)
x = r*cos(theta) = 150*cos(42) = 111.47172382161
the plane has traveled roughly 111 more miles eastward. Which adds onto the x coordinate of point B to get 32.16089+111.47172382161 = 143.63261382161 and that rounds to 144
This is the x coordinate of point C.
The plane has traveled about 144 miles to the east
Repeat for the y coordinate, but use sine instead of cosine
y = r*sin(theta)
y = 150*sin(42)
y = 100.369590953829
add this onto the y coord of point B
105.193523 + 100.369590953829 = 205.56311395383
which rounds to 206
This is the y coordinate of point C.
The plane has traveled about 206 miles to the north
Point C is approximately located at (x,y) = (143.6326, 205.5631)
If you rounded to the nearest whole number, then point C is roughly located at (144, 206)
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Answers:
Total distance traveled north: 206 miles
Total distance traveled east: 144 miles