This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Find side b using the Cosine Rule. The Cosine Rule states b^2 = a^2 + c^2 - 2ac B. Given a = 6.7 cm, c = 2.3 cm, and B = 46.6^. Substitute the values into the formula: b^2 = (6.7)^2 + (2.3)^2 - 2(6.7)(2.3) (46.6^) Calculate the squares: b^2 = 44.89 + 5.29 - 2(6.7)(2.3) (46.6^) Sum the squared terms: b^2 = 50.18 - 2(6.7)(2.3) (46.6^) Multiply the terms in the cosine part: b^2 = 50.18 - 30.82 (46.6^) Calculate (46.6^): (46.6^) ≈ 0.687004 Substitute this value: b^2 = 50.18 - 30.82(0.687004) b^2 = 50.18 - 21.179999999999996 b^2 = 29.000000000000004 Take the square root to find b: b = sqrt(29.000000000000004) b ≈ 5.38516 cm Rounding to three significant figures: b ≈ 5.39 cm Step 2: Find angle A using the Sine Rule. The Sine Rule states (a)/( A) = (b)/( B). We have a = 6.7 cm, b ≈ 5.38516 cm, and B = 46.6^. Rearrange the formula to solve for A: A = (a B)/(b) Substitute the values: A = (6.7 (46.6^))/(5.38516) Calculate (46.6^): (46.6^) ≈ 0.726591 Substitute this value: A = (6.7 × 0.726591)/(5.38516) A = (4.8681597)/(5.38516) A ≈ 0.90400 Now, find A by taking the inverse sine: A = (0.90400) A ≈ 64.690^ Rounding to one decimal place: A ≈ 64.7^ Step 3: Find angle C using the sum of angles in a triangle. The sum of angles in a triangle is 180^. A + B + C = 180^ We have A ≈ 64.690^ and B = 46.6^. Rearrange to solve for C: C = 180^ - A - B Substitute the values: C = 180^ - 64.690^ - 46.6^ C = 180^ - 111.290^ C = 68.710^ Rounding to one decimal place: C ≈ 68.7^ The final answers are: b ≈ 5.39 cm A ≈ 64.7^ C ≈ 68.7^