The Law of Cosines is a powerful tool that allows us to find unknown sides or angles in a triangle when we know certain other sides and angles. Unlike the Law of Sines, the Law of Cosines doesn't require us to have an angle and its opposite side. It's particularly useful when we know all three sides of a triangle (SSS) or two sides and the included angle (SAS). Understanding the Law of Cosines opens up a world of possibilities in geometry and trigonometry!
The Law of Cosines states:
$a^2 = b^2 + c^2 - 2bc \cdot cos(A)$
$b^2 = a^2 + c^2 - 2ac \cdot cos(B)$
$c^2 = a^2 + b^2 - 2ab \cdot cos(C)$
Match the terms with their correct definitions:
| Term | Definition |
|---|---|
| 1. Cosine | A. The side opposite the given angle. |
| 2. Angle | B. The trigonometric function adjacent/hypotenuse. |
| 3. Side | C. The figure formed by two rays sharing a common endpoint. |
| 4. Opposite | D. One of the line segments that form a polygon. |
| 5. Included Angle | E. The angle between two known sides in the Law of Cosines. |
Complete the following paragraph using the words: sides, angle, triangle, cosine, Law.
The _____ of Cosines is used to solve for missing _____ and _____ in a _____. This theorem uses the _______ function to relate the lengths of the sides to the _______ of one of its angles.
Explain in your own words why the Law of Cosines is a generalization of the Pythagorean Theorem.
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