Finding the angles of an isosceles triangle when you know all three sides is a straightforward process that utilizes the Law of Cosines. An isosceles triangle has two sides that are equal, which makes calculating the angles a bit easier.
Step 1: Identify the sides
Let’s label the sides of the triangle as follows: let the lengths of the equal sides be ‘a’ and the base side be ‘b’. So, we have:
- Side 1 = a
- Side 2 = a
- Base = b
Step 2: Use the Law of Cosines
The Law of Cosines states that for any triangle with sides a, b, and c opposite angles A, B, and C respectively, the following holds:
c² = a² + b² – 2ab * cos(C)
In our isosceles triangle, we need to find the angles opposite the sides. The angle opposite the base (b) is different from the angles opposite the equal sides (a). Let’s denote:
- Angle at the base = C
- Angles at the equal sides = A
To find angle C, which is opposite side b, we can rearrange the Law of Cosines:
cos(C) = (a² + a² – b²) / (2 * a * a)
This simplifies to:
cos(C) = (2a² – b²) / (2a²)
Now calculate C:
C = cos-1((2a² – b²) / (2a²))
Step 3: Calculate angles A
Since the triangle’s angles must sum up to 180 degrees, we can find angle A by:
A = (180 – C) / 2
Example
Consider an isosceles triangle with sides of length 5, 5, and 6. We can follow the steps:
1. Here, a = 5 and b = 6.
2. Calculate C:
cos(C) = (2 * 5² – 6²) / (2 * 5²) = (50 – 36) / 50 = 14 / 50 = 0.28
C = cos-1(0.28) ≈ 73.74°
3. Now calculate A:
A = (180 – 73.74) / 2 = 53.13°
Thus, the angles are approximately 53.13°, 53.13°, and 73.74°.
In summary, by using the Law of Cosines and the properties of isosceles triangles, you can efficiently determine the angles when you know the lengths of all three sides.