Precalc: Law of Cosines

Front

What is the Law of Cosines?

Back

The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. It is used to find a side or angle in a triangle when two sides and the included angle are known, or when all three sides are known.

Front

When is the Law of Cosines used?

Back

The Law of Cosines is used when you have either: 1) Two sides and the included angle (SAS) or 2) All three sides of a triangle (SSS).

Front

What is the formula for the Law of Cosines?

Back

c² = a² + b² - 2ab * cos(C), where a and b are the lengths of the sides, C is the included angle, and c is the length of the side opposite angle C.

Front

How do you find an angle using the Law of Cosines?

Back

To find an angle, rearrange the Law of Cosines formula: cos(C) = (a² + b² - c²) / (2ab). Then use the inverse cosine function.

Front

If a triangle has sides of lengths 7, 8, and 9, what is the angle opposite the side of length 9?

Back

Using the Law of Cosines: cos(C) = (7² + 8² - 9²) / (2 * 7 * 8) = 0.125. Therefore, C ≈ 82.82°.

Front

What is the relationship between the Law of Cosines and the Pythagorean theorem?

Back

The Law of Cosines generalizes the Pythagorean theorem. When the triangle is a right triangle, the Law of Cosines simplifies to the Pythagorean theorem.

Front

What does it mean if the Law of Cosines gives a negative value for cos(C)?

Back

If cos(C) is negative, it indicates that angle C is obtuse (greater than 90°).

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How can the Law of Cosines be applied in real-life situations?

Back

The Law of Cosines can be used in navigation, architecture, and various fields of engineering to determine distances and angles.

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What is the significance of the angle in the Law of Cosines?

Back

The angle in the Law of Cosines is crucial as it determines the relationship between the sides of the triangle. It affects the calculation of the lengths of the sides.

Front

If two sides of a triangle are 5 and 12, and the included angle is 60°, what is the length of the third side?

Back

Using the Law of Cosines: c² = 5² + 12² - 2 * 5 * 12 * cos(60°) = 169. Therefore, c = 13.