A ladder 8.5m long is placed with its foot at a distance of 4m from the wall of a house and just reaches a window will . find the height of a window will and the sine and tan of the angle which the ladder makes with the wall
To find the height of the window and the trigonometric ratios of the angle made by the ladder with the wall, we can use the concept of right triangles and trigonometric functions.
Let's denote the height of the window as "h" and the angle made by the ladder with the wall as "θ".
1. Height of the window (h):
The ladder, the distance of its foot from the wall (4m), and the height of the window (h) form a right triangle. The ladder acts as the hypotenuse of this triangle.
Using the Pythagorean theorem,
hypotenuse^2 = base^2 + height^2
ladder^2 = 4^2 + h^2
(8.5)^2 = 16 + h^2
h^2 = (8.5)^2 - 16
h^2 = 72.25 - 16
h^2 = 56.25
To find the value of h, take the square root of both sides:
h = √(56.25)
h ≈ 7.50 m (rounded to two decimal places)
Therefore, the height of the window is approximately 7.50 meters.
2. Trigonometric ratios:
The sine (sin) and tangent (tan) of the angle (θ) can be calculated using the sides of the right triangle.
- Sine (sin):
sin(θ) = opposite/hypotenuse
sin(θ) = h/ladder
sin(θ) = 7.50/8.5
sin(θ) ≈ 0.882 (rounded to three decimal places)
Therefore, the sine of the angle made by the ladder with the wall is approximately 0.882.
- Tangent (tan):
tan(θ) = opposite/adjacent
tan(θ) = h/4
tan(θ) = 7.50/4
tan(θ) ≈ 1.875 (rounded to three decimal places)
Therefore, the tangent of the angle made by the ladder with the wall is approximately 1.875.
In summary:
- The height of the window is approximately 7.50 meters.
- The sine of the angle made by the ladder with the wall is approximately 0.882.
- The tangent of the angle made by the ladder with the wall is approximately 1.875.
just good-old Pythagoras here.
y^2 + 4^2 = 8.5^2
...
y = √56.25
sinØ = √56.25/8.5 = ..
tanØ = √56.25/4 = ...