Longer leg: 4√5 cm,sin(θ): √(5/6),θ (radians): 0.9273,θ (degrees): 53.13°
Right Triangle Calculations
Identify the unknown side
We need to find the length of the longer leg (third side) and the sine ratio and measure of the larger acute angle.
Apply the Pythagorean theorem
Let x be the length of the longer leg. We know:
Hypotenuse (h) = 4√6 cm
Shorter leg (a) = 4√2 cm
Longer leg (x) = unknown
Using the Pythagorean theorem:
a^2 + b^2 = c^2
(4√2 cm)^2 + x^2 = (4√6 cm)^2
16 + x^2 = 96
x^2 = 80
x = √80 = 4√5 cm
Calculate the sine ratio of the larger acute angle
The sin ratio is the opposite side over the hypotenuse. Since the larger acute angle is opposite the longer leg, the sine ratio is:
sin(θ) = x / h = 4√5 cm / 4√6 cm
sin(θ) = √(5/6)
Calculate the measure of the larger acute angle in radians
We can use the inverse sine function (sin^-1) to find the angle from its sine ratio:
θ = sin^-1(√(5/6))
θ ≈ 0.9273 radians
Convert radians to degrees
There are 180 degrees in π radians, so:
θ in degrees = θ in radians * 180° / π
θ ≈ 0.9273 radians * 180° / π
θ ≈ 53.13 degrees (rounded to the hundredths place)