The angle theta is approximately 53.13 degrees
The length of the opposite side x is approximately 12.0 units.
How to solve for the angle θ and the length of the opposite side
To solve for the angle θ and the length of the opposite side (x), use trigonometric ratios in a right triangle.
In this case, we have the adjacent side (9), the hypotenuse (15), and we want to find the opposite side (x) and the angle θ.
The trigonometric ratio that relates the adjacent side and the hypotenuse is the cosine (cos) function:
cos(θ) = adjacent / hypotenuse
Substitute the known values:
cos(θ) = 9 / 15
To solve for θ, take the inverse cosine (arccos) of both sides:
θ = arccos(9 / 15)
Using a calculator to evaluate this expression, we find:
θ ≈ 53.13 degrees (rounded to two decimal places)
Therefore, the angle facing the opposite side is approximately 53.13 degrees.
To find the length of the opposite side (x), we can use the sine (sin) function:
sin(θ) = opposite / hypotenuse
Substituting the known values:
sin(θ) = x / 15
To solve for x, we can rearrange the equation:
x = 15 * sin(θ)
Using the value of θ we found earlier:
x ≈ 15 * sin(53.13 degrees)
Evaluating this expression:
x ≈ 12.0 (rounded to two decimal places)
Therefore, the length of the opposite side is approximately 11.33 units.