How to find sides of a right triangle if you have the hypotenuse of 20, alpha is 60 degrees and beta is 30 degrees

Answer 1

To find the sides of a right triangle given the hypotenuse and the measure of the other two angles, you can use trigonometric ratios.

Let's label the sides of the triangle:
- Hypotenuse: c = 20
- Opposite side to angle alpha (60 degrees): a
- Opposite side to angle beta (30 degrees): b

Now, we can use the sine, cosine, and tangent trigonometric ratios:

1. Sine (sin) ratio:
sin(alpha) = a / c
sin(60) = a / 20
√3/2 = a / 20
a = (√3/2) * 20
a = 10√3

2. Cosine (cos) ratio:
cos(alpha) = b / c
cos(60) = b / 20
1/2 = b / 20
b = (1/2) * 20
b = 10

3. Tangent (tan) ratio:
tan(beta) = a / b
tan(30) = a / 10
1/√3 = a / 10
a = (1/√3) * 10
a ≈ 10/√3
rationalize the denominator by multiplying the numerator and denominator by √3:
a ≈ (10/√3) * (√3/√3)
a ≈ (10√3) / 3

Therefore, the sides of the right triangle are approximately:
- Side opposite to angle alpha: a ≈ 10√3
- Side opposite to angle beta: b = 10
- Hypotenuse: c = 20