trig application problem
A ship is sailing due north. At a certain point the bearing of a lighthouse
that is 20 miles away is found to be 33 degrees east of due north. Later, the
bearing is determined to be 54 degrees east of due north. How far, to the
nearest tenth of a mile, has the ship traveled?
Somewhere ahead of the ship, a line drawn from the lighthouse to the course line
will form a right angle. That gives us a triangle with the hypotenuse equal to
the initial distance of 20 miles. We calculate the lengths of the other two legs,
which we will use later.
The length of the leg ahead of the ship is found by multiplying the sine
of 33 degrees by the length of the hypotenuse:
x = 20 miles * sin 33 = 20 miles * 0.5446 = 10.9 miles.
The length of the leg the ship is traveling on is found by multiplying
the cosine of 33 degrees by the length of the hypotenuse:
y = 20 miles * cos 33 = 20 miles * 0.8387 = 16.8 miles.
When the second bearing is taken, we need only calculate the remaining portion
of the leg the ship is on. In this new triangle, we already know the length of
the leg extending at a right angle out to the lighthouse: 10.89 miles. Divide
that by the tangent of the new bearing, 58 degrees, to find the distance to
the right angle ahead. Call this new distance y2.
y2 = 10.89 miles / tan 54 = 10.9 miles / 1.3764 = 7.9 miles
Subtract this remaining length from the original length of the course line:
16.8 miles - 7.9 miles = 8.9 miles
The ship has traveled 8.9 miles since the first bearing was taken.