A rectangle is 9.2m by 3.8m. What angle does its diagonal make with its longer side??
Tan x' = 3.8/9.2
Tan x' = 0.413
x' = 22.44' ( Degrees)
Oh, the diagonal of the rectangle is trying to make an angle? Well, it's like a rebellious teenager trying to make a statement! But let's focus on the math for a second. To find the angle, we can use the inverse tangent function, or arctan, if you want to get fancy. So the tangent of the angle is the ratio of the opposite side to the adjacent side. In this case, the opposite side is the height (3.8m) and the adjacent side is the width (9.2m). Plug those numbers into your calculator and ta-da! You'll have the angle.
To find the angle that the diagonal makes with the longer side of the rectangle, we can use trigonometry.
Step 1: Determine the length of the diagonal.
To find the length of the diagonal, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a^2 + b^2).
In this case, the longer side of the rectangle is the length of 9.2m, and the shorter side is the width of 3.8m. Therefore, we have a right triangle with sides of 9.2m, 3.8m, and the diagonal (c).
Using the Pythagorean theorem:
c^2 = 9.2^2 + 3.8^2
c^2 = 84.64 + 14.44
c^2 = 99.08
Taking the square root of both sides to find c:
c ≈ √99.08
c ≈ 9.95m
So, the length of the diagonal is approximately 9.95 meters.
Step 2: Calculate the angle.
To find the angle, we can use trigonometric functions. In this case, we will use the sine function.
The sine of an angle (θ) is equal to the length of the opposite side divided by the length of the hypotenuse. In our case, the longer side of the rectangle is the adjacent side, and the diagonal is the hypotenuse.
sin(θ) = opposite/hypotenuse
sin(θ) = 9.2/9.95
To find θ, we need to take the inverse sine (or arcsine) of both sides of the equation.
θ ≈ arcsin(9.2/9.95)
θ ≈ 60.28 degrees
So, the angle that the diagonal makes with the longer side of the rectangle is approximately 60.28 degrees.
To find the angle that the diagonal makes with the longer side of the rectangle, we can use trigonometry. Let's break down the steps:
Step 1: Draw a diagram of the rectangle and its diagonal. Label the length of the longer side (9.2m) as the base, and label the width of the rectangle (3.8m) as the height.
Step 2: Calculate the length of the diagonal using the Pythagorean theorem. The diagonal (d) of a rectangle can be found using the equation: d² = b² + h², where b is the base and h is the height. In this case, d² = 9.2² + 3.8².
Step 3: Square the values: 9.2² = 84.64 and 3.8² = 14.44.
Step 4: Add the squared values: d² = 84.64 + 14.44 = 99.08.
Step 5: Calculate the square root of the sum to find the length of the diagonal: d ≈ √99.08 ≈ 9.9531m.
Step 6: Now we have all three sides of a right-angled triangle: the base (9.2m), the height (3.8m), and the diagonal (9.9531m). We can use trigonometry to find the angle between the diagonal and the longer side.
Step 7: In a right-angled triangle, the tangent of an angle is equal to the ratio of the length of the opposite side to the adjacent side. Since we want the angle between the diagonal and the longer side, the angle we want to find is the inverse tangent (or arctan) of the ratio of the opposite side (height) to the adjacent side (base).
Step 8: Calculate the angle using the tangent inverse function (arctan) in a calculator or software. The angle ≈ arctan(3.8 / 9.2) ≈ 22.7 degrees.
Therefore, the angle that the diagonal makes with the longer side is approximately 22.7 degrees.