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 expressed as: c² = a² + b² - 2ab * cos(C), where C is the angle opposite side c.

Tags

CCSS.HSG.SRT.D.10

CCSS.HSG.SRT.D.11

Front

How do you use the Law of Cosines to find an angle?

Back

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

Tags

CCSS.HSG.SRT.D.10

CCSS.HSG.SRT.D.11

Front

What is the formula for the Law of Cosines?

Back

The formula is: c² = a² + b² - 2ab * cos(C). This can be rearranged to find any side or angle in a triangle.

Tags

CCSS.HSG.SRT.D.10

CCSS.HSG.SRT.D.11

Front

If a = 5, b = 7, and c = 10, what is angle C?

Back

Using the Law of Cosines: cos(C) = (5² + 7² - 10²) / (2 * 5 * 7) = -0.1. Therefore, angle C ≈ 95.74°.

Tags

CCSS.HSG.SRT.D.10

CCSS.HSG.SRT.D.11

Front

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

Back

The Law of Cosines generalizes the Pythagorean theorem. When angle C is 90°, the Law of Cosines simplifies to a² + b² = c².

Tags

CCSS.HSG.SRT.D.10

CCSS.HSG.SRT.D.11

Front

How can the Law of Cosines be used in real-world applications?

Back

The Law of Cosines can be used in navigation, architecture, and engineering to determine distances and angles in non-right triangles.

Tags

CCSS.HSG.SRT.D.10

CCSS.HSG.SRT.D.11

Front

What is the significance of the cosine function in the Law of Cosines?

Back

The cosine function relates the angle of a triangle to the lengths of its sides, allowing for the calculation of unknown angles or sides.

Tags

CCSS.HSG.SRT.D.10

CCSS.HSG.SRT.D.11

Front

If angle A = 60° and sides a = 8, b = 6, what is side c?

Back

Using the Law of Cosines: c² = 8² + 6² - 2 * 8 * 6 * cos(60°) = 64 + 36 - 48 = 52. Therefore, c = √52 ≈ 7.21.

Tags

CCSS.HSG.SRT.D.10

CCSS.HSG.SRT.D.11

Front

What is the Law of Cosines used for in triangle problems?

Back

It is used to find unknown sides or angles in any triangle, especially when the triangle is not a right triangle.

Tags

CCSS.HSG.SRT.D.10

CCSS.HSG.SRT.D.11

Front

How do you find the length of a side using the Law of Cosines?

Back

Rearrange the Law of Cosines formula to isolate the side you want to find. For example, to find side c: c = √(a² + b² - 2ab * cos(C)).

Tags

CCSS.HSG.SRT.D.10

CCSS.HSG.SRT.D.11