Given a right triangle with legs 8 and 5, find the measure of the two acute angles. (Round to the nearest hundredth
To find the measure of the acute angles in a right triangle, we can use trigonometric ratios.
The two main trigonometric ratios we can use for right triangles are sine (sin) and cosine (cos). In this case, we will use the sine ratio.
The sine of an angle is equal to the length of the side opposite the angle divided by the length of the hypotenuse. In this case, let's call the two acute angles A and B.
The sine of angle A will be equal to the length of the side opposite angle A (8) divided by the length of the hypotenuse. The hypotenuse can be found using the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the two legs is equal to the square of the hypotenuse.
Using the Pythagorean theorem, we can find the length of the hypotenuse:
Hypotenuse^2 = leg1^2 + leg2^2
Hypotenuse^2 = 8^2 + 5^2
Hypotenuse^2 = 64 + 25
Hypotenuse^2 = 89
Now, let's solve for the length of the hypotenuse:
Hypotenuse = √89
Now, let's find the sine of angle A:
sin(A) = opposite/hypotenuse
sin(A) = 8/√89
Using a calculator, compute the value of sin(A), and then take the inverse sine (also called arcsine) to find the measure of angle A.
sin(A) ≈ 0.8571
A ≈ sin^(-1)(0.8571)
A ≈ 59.04 degrees
Similarly, we can find the measure of angle B using the sine ratio:
sin(B) = opposite/hypotenuse
sin(B) = 5/√89
Using a calculator, compute the value of sin(B), and then take the inverse sine to find the measure of angle B.
sin(B) ≈ 0.5351
B ≈ sin^(-1)(0.5351)
B ≈ 32.59 degrees
Therefore, the measure of the two acute angles in this right triangle, rounded to the nearest hundredth, is approximately 59.04 degrees and 32.59 degrees.