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.
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Another one Wamandu — let's solve it. Trigonometry has several laws that help us solve triangles. The two most fundamental are the Law of Sines and the Law of Cosines. Law of Sines This law relates the lengths of the sides of a triangle to the sines of its opposite angles. For a triangle with sides a, b, c and opposite angles A, B, C, the law states: (a)/(sin A) = (b)/(sin B) = (c)/(sin C) Application: The Law of Sines is useful when you know: • Two angles and one side (AAS or ASA). • Two sides and an angle opposite one of them (SSA — this case can sometimes have two possible solutions, known as the ambiguous case). Example: Consider a triangle where angle A = 30°, angle B = 45°, and side a = 10. We can find side b using the Law of Sines: Step 1: Set up the proportion: (a)/(sin A) = (b)/(sin B) (10)/(sin 30°) = (b)/(sin 45°) Step 2: Substitute known sine values (sin 30° = 0.5, sin 45° = √(2)/2 ≈ 0.707): (10)/(0.5) = (b)/(√(2)/2) Step 3: Solve for b: 20 = (b)/(√(2)/2) b = 20 × (√(2)/2) b = 10√2 Law of Cosines This law is an extension of the Pythagorean theorem and relates the lengths of the sides of a triangle to the cosine of one of its angles. It states: c² = a² + b² - 2ab cos C a² = b² + c² - 2bc cos A b² = a² + c² - 2ac cos B Application: The Law of Cosines is useful when you know: • All three sides (SSS). • Two sides and the included angle (SAS). Example: Consider a triangle where side a = 7, side b = 8, and angle C = 60°. We can find side c using the Law of Cosines: Step 1: Use the formula: c² = a² + b² - 2ab cos C c² = 7² + 8² - 2(7)(8) cos 60° Step 2: Substitute known values (cos 60° = 0.5): c² = 49 + 64 - 2(56)(0.5) c² = 113 - 56 Step 3: Solve for c: c² = 57 c = √57 Send me the next one 📸