From a point A, due south of a tower , the angle of elevation of the top is 45degree, another point B is due east of the tower and the bearing of B from A is 30degree. Find the angle of elevation of the top of the tower B
Label the top of the tower T and the bottom S.
If the height ST is h, then you know that SA = h
since angle SAB is 30°, SB = h/√3
Now you have triangle SBT, which is congruent to SAB.
so, ...
To find the angle of elevation of the top of the tower from point B, we can use a trigonometric approach.
First, let's visualize the scenario. Point A is due south of the tower, so let's assume the tower is at the origin (0,0) on a coordinate plane. Point A would then have coordinates (0,-x), where x is the distance from point A to the tower.
Since the angle of elevation of the top of the tower from point A is 45 degrees, we can draw a right triangle with one side on the ground (the line connecting point A to the tower) and the other side being the line connecting the top of the tower to the ground. The angle between these two sides is 45 degrees.
Now, let's find the length of the side opposite to the 45-degree angle in this right triangle. This will be the height of the tower above the ground.
Using trigonometry, we know that the tangent of an angle in a right triangle is equal to the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.
In this case, the tangent of 45 degrees is equal to the height of the tower (opposite side) divided by the distance from point A to the tower (adjacent side).
Since the tangent of 45 degrees is 1, we have:
1 = height of the tower / x
Simplifying, we find that the height of the tower is equal to x.
Now, let's move on to point B. We know that point B is due east of the tower, and the bearing of B from A is 30 degrees.
The bearing of B from A refers to the angle between the line connecting point B to the tower and the north line. In other words, the bearing is the angle measured clockwise from the north line to the line connecting the two points.
Since B is due east of the tower, the bearing from A is 90 degrees. Adding the clockwise angle of 30 degrees, we find that the bearing of B from A is 120 degrees.
Now, let's find the angle of elevation of the top of the tower from point B.
Since the tower is at the origin (0,0) and point B is due east of the tower, the horizontal distance from the tower to point B is x.
Using trigonometry again, we can find the angle of elevation from point B.
The tangent of the angle of elevation is equal to the height of the tower (opposite side) divided by the distance from the tower to point B (adjacent side).
Since we already found that the height of the tower is x and the horizontal distance from the tower to point B is also x, we have:
tan(angle of elevation from B) = x / x
Simplifying, we find that the tangent of the angle of elevation from B is equal to 1.
To find the angle of elevation from B, we need to determine the inverse tangent (arctan) of 1.
arctan(1) ≈ 45 degrees
Therefore, the angle of elevation of the top of the tower from point B is approximately 45 degrees.