Law of Cosines Flashcard

Front

What is the Law of Cosines?

Back

The Law of Cosines states that in any triangle, the square of the length of one side is equal to the sum of the squares of the lengths of the other two sides minus twice the product of the lengths of the other two sides multiplied by the cosine of the included angle. It is expressed as: c² = a² + b² - 2ab * cos(C).

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 to find the angle.

Front

What is the relationship between the sides and angles in a triangle?

Back

In a triangle, the larger the angle, the longer the opposite side. Conversely, the smaller the angle, the shorter the opposite side.

Front

What is the formula for finding the length of a side using the Law of Cosines?

Back

The formula for finding the length of a side using the Law of Cosines is: c² = a² + b² - 2ab * cos(C), where c is the side opposite angle C.

Front

If a triangle has sides of lengths 7, 8, and 9, how can you find the angles?

Back

Use the Law of Cosines. For example, to find angle C: cos(C) = (7² + 8² - 9²) / (2 * 7 * 8). Calculate the cosine and then use the inverse cosine to find angle C.

Front

What is the significance of the Law of Cosines in solving triangles?

Back

The Law of Cosines is significant because it allows for the calculation of unknown sides or angles in any triangle, not just right triangles.

Front

How can the Law of Cosines be applied in real-world problems?

Back

The Law of Cosines can be applied in various fields such as engineering, architecture, and navigation to determine distances and angles when direct measurement is not possible.

Front

What is the difference between the Law of Cosines and the Law of Sines?

Back

The Law of Cosines is used when you know two sides and the included angle or all three sides, while the Law of Sines is used when you know two angles and a side or two sides and a non-included angle.

Front

What is the formula for the Law of Sines?

Back

The Law of Sines states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant. It is expressed as: a/sin(A) = b/sin(B) = c/sin(C).

Front

How do you determine if a triangle is valid using side lengths?

Back

A triangle is valid if the sum of the lengths of any two sides is greater than the length of the third side (Triangle Inequality Theorem).