Question

a man measured the angle of elevation of the too of a tower to be 70°.when he walked 30m further,the angle of elevation of the top of the tower was 35°.find yhe diatance from the top of the tower to the second observation point.

Answer 1

Did you make a sketch?
Look at the triangle formed by the two observation points and the top of
the tower. We know all the angles, and we have a side, so we can use
the sine law.
Let the distance you want be x
x/sin110° = 30/sin35°
x = 30sin110/sin35 = appr 49.15 m

Check by doing it another way:
notice the triangle is also isosceles, so by the cosine law:
x^2 = 30^2 + 30^2 - 2(30)(30)cos110
= 900+900-1800cos110
= 2415.63...
x = 49.15 m, same as above

Answer 2

or, starting with the usual problem of finding the height (h) of the tower, you could do
h cot35° - h cot70° = 30
h = 30/(cot35° - cot70°) = 28.19
Now the desired distance (x) is
h/x = sin35°
x = h/sin35° = 49.15 as above