The length of the shadows of a vertical pole of height h , thrown by the sun's rays at three different moments are h , 2 h and 3 h . The sum of the angles of elevation of the rays at these three moments is equal to

Step-by-step solution

Let O A be the vertical pole of height h and O P 1 , O P 2 , O P 3 be the lengths of shadow. In ∆ A O P 1 we have, tan ⁡ θ 1 = O A O P 1 = h h = 1 ⟹ θ 1 = π 4 In ∆ A O P 2 we have, tan ⁡ θ 2 = O A O P 2 = 2 2 h = 1 2 ⟹ θ 2 = tan - 1 ⁡ 1 2 Similarly, tan ⁡ θ 3 = O A O P 3 = h 3 h = 1 3 θ 3 = tan - 1 ⁡ 1 3 ∴ Sum of the angles = θ 1 + θ 2 + θ 3 = π 4 + tan - 1 ⁡ 1 2 + tan - 1 ⁡ 1 3 = π 4 + tan - 1 ⁡ 1 2 + 1 3 1 - 1 2 × 1 3 = π 4 + tan - 1 ⁡ 1 = π 4 + π 4 = π 2