Find the value of x using trigonometric ratios.
24-32 right triangle
show work
In a 24-32 right triangle, the ratio of the two legs is 3:4, which implies that the other side must be in the ratio 3:4:5 (Pythagorean triples). We can test this by finding the square of the larger leg, 32^2 = 1024, and adding the square of the smaller leg, 24^2 = 576. The sum is 1024 + 576 = 1600.
Taking the square root of this sum, we get sqrt(1600) = 40, which is indeed 5 times the length of the smaller leg. So the sides of the triangle are 24, 32, and 40.
As for the value of x, it depends on which angle in the triangle we are referring to. If we consider the angle opposite to the side of length 24, then we can define x as the sine, cosine, or tangent of that angle.
- sine (opposite/hypotenuse): sin(x) = 24/40 = 3/5
- cosine (adjacent/hypotenuse): cos(x) = 32/40 = 4/5
- tangent (opposite/adjacent): tan(x) = 24/32 = 3/4
Depending on which trigonometric ratio is given or requested, the value of x will be the angle that yields the corresponding ratio. Using a calculator or a trigonometric table:
- sin(x) = 3/5: x ≈ 36.87°
- cos(x) = 4/5: x ≈ 53.13°
- tan(x) = 3/4: x ≈ 36.87° (same as sine since it's the opposite angle)
This means that the two acute angles in the triangle are approximately 36.87° and 53.13° (which sum to 90°, as expected).