Relations Between Sides and Angles of a Triangle

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The Sine Rule (or Sine Formula) states: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ Where: - a, b, c are the sides of the triangle - A, B, C are the angles opposite to sides a, b, c r…

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Step 1: Identify the given information - a = 8 cm, ∠A = 60°, ∠B = 45° - Need to find: b Step 2: Apply Sine Rule $$\frac{a}{\sin A} = \frac{b}{\sin B}$$ Step 3: Substitute values $$\frac{8}{\sin 60°}…

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Step 1: Identify what's given - a = 5 cm, b = 7 cm, ∠A = 30° - Need to find: ∠C Step 2: First, find ∠B using Sine Rule $$\frac{a}{\sin A} = \frac{b}{\sin B}$$ $$\frac{5}{\sin 30°} = \frac{7}{\sin B}$…

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The Cosine Rule (or Cosine Formula) states: $$\cos A = \frac{b^2 + c^2 - a^2}{2bc}$$ $$\cos B = \frac{c^2 + a^2 - b^2}{2ac}$$ $$\cos C = \frac{a^2 + b^2 - c^2}{2ab}$$ Alternatively, the rule can be …

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Step 1: Identify the largest side - The largest side is c = 7 cm - The angle opposite to the largest side is the greatest angle - Therefore, ∠C is the greatest angle Step 2: Apply Cosine Rule for ∠C …

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Step 1: Find angle A using Cosine Rule $$\cos A = \frac{b^2 + c^2 - a^2}{2bc} = \frac{9 + 16 - 4}{2(3)(4)} = \frac{21}{24} = \frac{7}{8}$$ $$A = \arccos(\frac{7}{8}) ≈ 28.96°$$ Step 2: Find angle B $…

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Step 1: Identify what's given - b = 5 cm, c = 7 cm, ∠A = 60° - Need to find: side a - This is SAS (Side-Angle-Side) case Step 2: Use Cosine Rule in the form $$a^2 = b^2 + c^2 - 2bc \cos A$$ Step 3: …

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The Projection Formula states: $$a = b\cos C + c\cos B$$ $$b = c\cos A + a\cos C$$ $$c = a\cos B + b\cos A$$ What does this mean? - Each side of a triangle equals the sum of the projections of the o…