Approximating the Acute Angles of a Triangle with Sides 5, 12, and 13
We are given a triangle with sides of length 5, 12, and 13. Notice that these lengths form a Pythagorean triple (5² + 12² = 13²), indicating that this is a right triangle.
Let's label the angles:
- A is the angle opposite the side with length 5.
- B is the angle opposite the side with length 12.
- C is the angle opposite the side with length 13 (the hypotenuse).
Since it's a right triangle, we know angle C is 90 degrees. To approximate the acute angles A and B, we can use trigonometric ratios:
Finding Angle A
- Sine (sin): sin(A) = opposite side / hypotenuse = 5/13
- Cosine (cos): cos(A) = adjacent side / hypotenuse = 12/13
- Tangent (tan): tan(A) = opposite side / adjacent side = 5/12
We can use any of these ratios to find angle A. Let's use the sine function:
sin(A) = 5/13
To find angle A, we need to use the inverse sine function (arcsin):
A = arcsin(5/13) ≈ 22.62 degrees
Finding Angle B
Similarly, we can use any trigonometric ratio to find angle B. Let's use the tangent function:
tan(B) = 12/5
B = arctan(12/5) ≈ 67.38 degrees
Therefore, the approximate acute angles of the triangle are:
- Angle A ≈ 22.62 degrees
- Angle B ≈ 67.38 degrees
