give me the solution from the top of a building the angle of depression of a point on the same horizontal plane with the base of the building is observed to be 60 degree what will be the angle of depression of the same point when viewed from a position one third of the way up the building
if the building's height is h,
the distance to P, call it x,
tan60° = h/x
x = h/√3
from h/3, the angle θ has
tanθ = (h/3)/(h/√3) = 1/√3
θ = 30°
Let's break down the problem step by step:
Step 1: We are given that the angle of depression from the top of the building to a point on the same horizontal plane with the base is 60 degrees.
Step 2: When viewing the same point from a position one-third of the way up the building, we want to find the new angle of depression.
Step 3: Let's label the distance from the top of the building to the point on the horizontal plane as 'd'. Since we know the angle of depression is 60 degrees, we can determine the height of the building using the trigonometric function tangent.
tan(60) = height of the building / d
Simplifying, we have:
√3 = height of the building / d
Step 4: Now, let's determine the new angle of depression from a position one-third of the way up the building. Since the position is one-third of the way up, the distance from this position to the point on the horizontal plane is (2/3)d.
Step 5: Let's label the new angle of depression as 'x'. Using the same trigonometric function, tangent, we can determine the new height of the building.
tan(x) = height of the building / (2/3)d
Step 6: To find the value of 'x', we can substitute the expression for the height of the building (from Step 3) into the equation from Step 5:
tan(x) = (√3)d / (2/3)d
Simplifying, we get:
tan(x) = (√3)(3/2)
Step 7: Now we can solve for 'x' by taking the inverse tangent (arctan) of both sides:
x = arctan(√3)(3/2)
x ≈ 48.19 degrees
Therefore, the angle of depression of the same point when viewed from a position one-third of the way up the building is approximately 48.19 degrees.
To find the angle of depression at a different position on the building, we need to understand the concept of trigonometry. The angle of depression is the angle formed between the line of sight from the observer to the point and the horizontal line.
Let's break down the problem step by step:
1. We know that the angle of depression from the top of the building to a point on the same horizontal plane is 60 degrees.
2. To find the angle of depression from a position one third of the way up the building, we need to calculate the height of the building.
3. Let's assume that the height of the building is 'h' units.
4. From the given information, we know that the vertical distance from the top of the building to the observed point (on the same horizontal plane) is 'h' units.
5. Using trigonometry, we can say that the height of the building 'h' is equal to the tangent of the angle of depression (60 degrees) multiplied by the horizontal distance between the observer and the base of the building.
So, h = tan(60 degrees) * x .......(1), where x is the horizontal distance between the observer and the base of the building.
6. Now, we need to find the angle of depression from a position one third of the way up the building. Let's say the height from this position to the observed point is 'h1' units.
7. Using a similar trigonometric relationship, we can say that h1 = tan(angle of depression) * (2/3) * h .......(2), where (2/3)h is the vertical distance from the top of the building to the observer at one third of the building's height.
8. We can substitute equation (1) into equation (2):
h1 = tan(angle of depression) * (2/3) * (tan(60 degrees) * x)
Simplifying this equation gives us the required expression.
Therefore, to find the angle of depression from a position one-third of the way up the building, we need to substitute the values of 'angle of depression' and 'x' into the equation above.