Find the measures of the acute angles of a right triangle whose legs are 9 cm and
16 cm long.
or, just find one, and the other is its complement
use tan x=9/16 and tan y=16/9 to find the two angle x and y,then u will get the ans
To find the measures of the acute angles of a right triangle, we can use the trigonometric ratios - sine, cosine, and tangent.
Given that the legs of the right triangle are 9 cm and 16 cm long, we can label the shorter leg (opposite the acute angle we want to find) as "a" and the longer leg (adjacent to the acute angle) as "b".
Let's find the acute angle opposite side "a" using the sine ratio:
sin(A) = a / hypotenuse
In this case, the hypotenuse is the longest side, which we can find using the Pythagorean Theorem:
c² = a² + b²
c² = 9² + 16²
c² = 81 + 256
c² = 337
c ≈ √337 (approximately 18.356)
Now, we can find the sine of angle A:
sin(A) = a / c
sin(A) = 9 / 18.356
sin(A) ≈ 0.489
To find the angle measure, we can take the inverse sine (sine⁻¹) of 0.489:
A ≈ sin⁻¹(0.489)
A ≈ 29.628°
Since we know that the sum of the acute angles in a right triangle is always 90°, we can find the measure of the other acute angle by subtracting the known angle from 90°:
Other acute angle ≈ 90° - 29.628°
Other acute angle ≈ 60.372°
Therefore, the measures of the acute angles in the right triangle are approximately 29.628° and 60.372°.
To find the measures of the acute angles of a right triangle, we can use trigonometric ratios. In this case, we have the lengths of the legs of the right triangle, which are 9 cm and 16 cm.
The trigonometric ratio we can use is tangent (tan). The tangent of an angle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.
Let's denote the angle opposite the leg of length 9 cm as angle A, and the angle opposite the leg of length 16 cm as angle B.
From the given information, we can write the following proportion:
tan(A) = 9/16
To find the measure of angle A, we can find the inverse tangent (also called arc tangent or atan) of both sides of the equation:
A = atan(9/16)
Using a calculator, perform the inverse tangent of 9/16:
A ≈ 29.74 degrees
Similarly, to find the measure of angle B, we can use the following proportion:
tan(B) = 16/9
B = atan(16/9)
Using a calculator, perform the inverse tangent of 16/9:
B ≈ 60.96 degrees
Therefore, the measures of the acute angles of the right triangle with legs of lengths 9 cm and 16 cm are approximately 29.74 degrees and 60.96 degrees, respectively.