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The hypotenuse of a right triangle measures 4√6 centimeters and its shorter leg measures 4√2 centimeters. What is the measure of the 3rd side? What is the sin ratio of the larger acute angle? What is the measure of the large acute angle in radians? What is the measure of the large acute angle in degrees rounded to the hundredths place?

User Apomene
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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)

User Dale Ragan
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