Cosine rule
Cosine Rule
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The Cosine rule is an important rule in trigonometry often useful in situations where the Pythagoras’ theorem and sine rule can’t be applied, especially when there is no mention of a right angle.
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The rule is primarily used to find an unknown side or an angle in any triangle, specifically when we have either three sides or two sides and the included angle known.
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It is worth noting that the Cosine rule is a flexible method that can be used in any triangle, not just in right-angled triangles.
Formulation of Cosine Rule
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In a triangle with sides lengths
a,bandcand the angle opposite sidecisC, the Cosine rule can be expressed as: c² = a² + b² - 2ab cos(C). -
When rearranged to find an angle, the Cosine rule becomes: cos(C) = (a² + b² - c²) / (2ab).
Practical Usages
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The Cosine rule, like many mathematical concepts, has real-world applications in various professions including surveying, architecture, physics, engineering, and computer graphics.
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In these fields, the rule is used to calculate unknown distances or angles, helping to create accurate measurements and designs.
Solving Problems
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To find an unknown side, input the lengths of the known sides and the included angle into the Cosine rule and solve for the unknown.
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To find an unknown angle, input the lengths of all sides into the rearranged Cosine rule formula and compute the arccos (inverse cosine) of the calculated value.
Remember, the rule can be quite tricky to remember. Practice enough problems to get comfortable with the Cosine rule. Consistent practice will make it easier to understand and apply it in solving geometry problems.