Some Applications of Trigonometry

Angle of elevation is formed when we look upward, Angle of depression is formed when we look downward, Both angles are measured from the horizontal

Angle of elevation occurs when looking above horizontal level, angle of depression when looking below. Both are measured from the horizontal reference line. There's no fixed relationship between their magnitudes, and the line of sight changes direction based on the viewing angle.

50√3 meters

Using tan θ = opposite/adjacent. For angle of depression, tan 30° = 50/distance. Since tan 30° = 1/√3, we get distance = 50/(1/√3) = 50√3 meters.

3/4

If sin θ = 3/5, then opposite = 3 and hypotenuse = 5. Using Pythagoras theorem: adjacent = √(5² - 3²) = √(25 - 9) = √16 = 4. Therefore, tan θ = opposite/adjacent = 3/4.

tan⁻¹(3/2) and tan⁻¹(2/3)

From 4m: tan α = 6/4 = 3/2, so α = tan⁻¹(3/2). From 9m: tan β = 6/9 = 2/3, so β = tan⁻¹(2/3). Since tan α × tan β = (3/2) × (2/3) = 1, the angles are complementary (α + β = 90°).