How to Solve the Triangle with Sides b = 36, a = 42, c = 18?

To solve the triangle with the given sides, we can use the Law of Cosines. The Law of Cosines states that for any triangle with sides a, b, and c, the following relationships hold:

c² = a² + b² – 2ab * cos(C)

In our case, we need to solve for the angles given sides a = 42, b = 36, and c = 18. First, let’s find angle C:

Substituting the values into the equation:

18² = 42² + 36² – 2 * 42 * 36 * cos(C)

Calculating the squares:

324 = 1764 + 1296 – 3024 * cos(C)

Now, simplify:

324 = 3060 – 3024 * cos(C)

Rearranging gives us:

3024 * cos(C) = 3060 – 324

3024 * cos(C) = 2736

Now, solving for cos(C):

cos(C) = 2736 / 3024

Calculating this gives:

cos(C) ≈ 0.904

To find angle C, we take the inverse cosine:

C ≈ cos^-1(0.904) ≈ 25.84°

Next, we can find angle A using the Law of Sines:

sin(A) / a = sin(C) / c

This gives:

sin(A) / 42 = sin(25.84°) / 18

Thus:

sin(A) = 42 * sin(25.84°) / 18

Calculating sin(25.84°) gives approximately 0.437, so:

sin(A) ≈ 42 * 0.437 / 18 ≈ 0.918

Using the inverse sine:

A ≈ sin^-1(0.918) ≈ 66.56°

Finally, we can find angle B by recognizing that the sum of the angles in a triangle equals 180°:

B = 180° – A – C

B = 180° – 66.56° – 25.84° ≈ 87.6°

In conclusion, the angles of the triangle are approximately:

• Angle A: 66.56°
• Angle B: 87.6°
• Angle C: 25.84°