Law of Cosines Derivation/Proof
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8 months ago
Published on Apr 22, 2024
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Table of Contents
Step-by-Step Tutorial: Derivation of the Law of Cosines
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Introduction to the Law of Cosines:
- The Law of Cosines allows us to find the length of the third side of a triangle when we know the lengths of two sides and the angle between them.
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Setup of the Triangle:
- Consider a triangle with sides labeled as A, B, and C, and an angle labeled as Theta.
- Draw a perpendicular line from the vertex of angle Theta to side B, creating a right triangle.
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Expressing Side Lengths in Terms of Other Variables:
- Define side lengths A and C in terms of the perpendicular height (H) and express side B in terms of sides A and C.
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Calculating Side Lengths:
- Calculate side x using the cosine of Theta: x = A * cos(Theta).
- Express side B in terms of A and Theta: B = A - A * cos(Theta).
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Applying the Pythagorean Theorem:
- Apply the Pythagorean theorem to the right triangles in the setup:
- (A^2 = x^2 + H^2)
- (C^2 = H^2 + (B - A*cos(Theta))^2)
- Apply the Pythagorean theorem to the right triangles in the setup:
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Simplifying Equations:
- Simplify the equations obtained from the Pythagorean theorem by expanding and simplifying terms.
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Substitution and Simplification:
- Substitute (A^2 = x^2 + H^2) into the second equation to simplify it further.
- Simplify the expressions to obtain (C^2 = A^2 + B^2 - 2Acos(Theta)*B).
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Finalizing the Law of Cosines:
- Rearrange the terms in the simplified equation to obtain the traditional form of the Law of Cosines: (C^2 = A^2 + B^2 - 2AB*cos(Theta)).
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Conclusion:
- Understanding the derivation of the Law of Cosines helps in solving problems related to triangles where the side lengths and angles are known.
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Further Learning:
- Explore more videos and resources to deepen your understanding of trigonometry and geometric principles.
By following these steps, you can derive and understand the Law of Cosines, a fundamental concept in trigonometry.