Angle of elevation and depression of a point

Suppose a straight line AX is drawn in the horizontal direction. Then the angle XAP where P is a point above AX is called the angle of elevation of P as seen from A. similarly the angle XAQ where Q is below AX, is called the angle of depression of some point Q.

Line perpendicular to a plane is perpendicular to energy line lying in the plane.

To express one side of a right angled triangle in terms of the other side.

Let < ABC = θ, where ABC is a right angled triangle in which < C = 90

The side opposite right angle C will be denoted by H (hypotenuse), the side opposite to angle 4:07 PM will be denoted by o (opposite) and the side containing the angle θ (Other than H) will be denoted by A i.e., adjacent side.

Then from the figure it is clear that

O = A (tan θ) or A = O (cot θ)

I.e., opposite = Adj tan θ

Also O = H (sin θ or A = H (cos θ

Or opposite = Hyp sin θ

Or Adj = Hyp (cos θ)

Geometrical properties for a triangle:

In a triangle the internal bisector of an angle divides the opposite side in the ratio of the arms of the angle.

BD/DC = c/b

In an isosceles triangle the median is perpendicular to the base.

In similar triangles the corresponding sides are proportional.

The exterior angle is equal to sum of interior opposite angles.

θ= A + B

Example:

A ladder rests against a wall at an angle X to the horizontal its foot is pulled away from the wall through a distance a so that it slides a distance b down the wall making an angle \(\beta \) with the horizontal, show that

a = b tan [(α + β)/2]

Solution:

b = BC = AB – AC = lsin α – lsin β

a = PQ = AQ – AP = lcos β – lcos α

a/b = cosα – cosβ / sinβ – sinα = tan [(α + β)/2]