Trig: Law of Sines?

Question

Find the Height of the Airplane:

An airplane is spotted by two observers who are 1000 feet apart. As the airplane passes over the line joining them, each observer takes a sighting of the angle of elevation to the plane. One observes 40degrees, other 35degrees. How high is the airplane?

Answer 1

Draw triangle ABC. AB is on the ground and 1000 ft, C is in the air and corresponds to the airplane. Angle A is 40 and angle B is 35. Now drop a perpendicular from C to the ground at point D. Angle C is 105, and the pdlr divides it into angles of 50 and 55 degrees. By the law of sines we can show:
sin 50/sin 55 = AD/BD.
Since AD+BD= 1000, we can find either length.
Suppose we use AD.
Then AD/sin 50 = CD/sin 40. Find CD.

Answer 2

first draw a triangle with the long side on the ground. The two angles given are 35 and 40 degrees. the final angle must be 105 degrees since all the angles in a triangle add up to 180.

Then you use the law of sines to say [sin(105)/1000]=[sin(35)/x] when x is the length of the side opposite 35 degrees. When you solve for x you get 593.81 feet.

Then you split the large triangle into two, dropping a perpindicular line from the large angle to the large side. This gives you a triangle with angles 40-50-90 and hypot. of x. you can then solve for the height by using h=[x{sin(x)}], which comes out to a height of 381.69 feet.

Answer 3

First draw a diagram
Call the observers A and B and the plane C
In this triangle AB = 1000 and m The sum of the angles of a triangle is 180 degrees so
M ___c___ = ___a____
sin C ...............Sin A

1000 = BC
-------- -------
sin 105 sin 40

BC = 1000(.6428)/(.9659) or about 665.5

Now Draw a line from C perpendicular to AB, this is the height

Notice we have a right triangle and Sin 35 = height/BC
but we just found BC so.... .5736 *665.5 = h
or h is about 381.7 feet off the ground.

Answer 4

if the plane is y feet high, then
assuming that when the observation is made, the plane is at some point vertically between the observers, as against a possibility to being on the side of one of the observers...

y/tan40 + y/tan35 = 1000
so
y=1000/(cot40 + cot35)
or
y=381.69 feet

Answer 5

the main important perspective in a triangle is often opposite of the longest component, that's b thus. regulation of cosines states that: a^2 = b^2 + c^2 - 2bc * cos A. that's such as : b^2 = a^2 + c^2 - 2ac * cos B. So, put in each and every of the given variables: 10^2 = 6^2 + 7^2 - 2(6)(7) * cos B. simplify - 10^2 = 6^2 + 7^2 - 2(6)(7) * cos B. one hundred = 36 + 40 9 - 80 4(cos B) one hundred = 85 - 80 4(cos B) 80 4(cos B) = -15 cos B = -15/80 4 = -5/28 Use inverse cosine for degree of perspective B - cos-a million (-5/28) = approximately one hundred.28656.