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Questions and Answers
Which formula relates the lengths of the sides of a triangle to the cosine of one of its angles?
Which formula relates the lengths of the sides of a triangle to the cosine of one of its angles?
What is the formula for the law of cosines?
What is the formula for the law of cosines?
When does the law of cosines reduce to the Pythagorean theorem?
When does the law of cosines reduce to the Pythagorean theorem?
What is the cosine of a right angle?
What is the cosine of a right angle?
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What does the law of cosines generalize?
What does the law of cosines generalize?
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Which triangle does the law of cosines apply to?
Which triangle does the law of cosines apply to?
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What does the law of cosines relate?
What does the law of cosines relate?
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Which angle does the law of cosines use?
Which angle does the law of cosines use?
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What happens if the triangle is a right triangle?
What happens if the triangle is a right triangle?
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What is the formula for the law of cosines?
What is the formula for the law of cosines?
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Study Notes
Law of Cosines Overview
- Relates the lengths of the sides of a triangle to the cosine of one of its angles.
- Formula: ( c^2 = a^2 + b^2 - 2ab \cdot \cos(C) ) where ( C ) is the angle opposite side ( c ).
Reduction to Pythagorean Theorem
- The law of cosines simplifies to the Pythagorean theorem when the triangle is a right triangle (i.e., ( C = 90^\circ )).
- In a right triangle, the formula becomes ( c^2 = a^2 + b^2 ) since ( \cos(90^\circ) = 0 ).
Cosine and Right Angles
- The cosine of a right angle is 0.
- This is a crucial point for simplifying the law of cosines in right triangles.
Generalization of the Law of Cosines
- Generalizes the Pythagorean theorem for any type of triangle, not just right triangles.
- Applicable to obtuse, acute, and right triangles.
Applicability of the Law of Cosines
- The law applies to any triangle, regardless of its angle sizes.
- It allows for calculations of side lengths and angles that are not determinable using the Pythagorean theorem alone.
Relationships and Uses
- Relates the lengths of the sides of a triangle with the cosine of one of its angles, allowing for the calculation of unknown side lengths or angles.
- Useful in various geometric and trigonometric problems.
Summary of Law of Cosines Formula
- Formula representation: ( c^2 = a^2 + b^2 - 2ab \cdot \cos(C) ).
- Can be adapted for other angles by appropriately rearranging or substituting sides and angles involved.
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Description
Test your knowledge of the law of cosines with this quiz! Explore how the lengths of triangle sides relate to the cosine of its angles. Challenge yourself with questions about the formula and its applications.
