Review Flashcard #4 Law of Cosines and area of trangles

Front

What is the Law of Cosines?

Back

The Law of Cosines states that for any triangle with sides a, b, and c, and angles A, B, and C opposite those sides respectively, the following holds: c² = a² + b² - 2ab * cos(C). It is used to find a side or angle in a triangle when the other sides and angles are known.

Front

When do you use the Law of Cosines?

Back

The Law of Cosines is used when you know two sides and the included angle (SAS) or all three sides (SSS) of a triangle.

Front

What is the formula for finding the area of a triangle using the Law of Cosines?

Back

Area = 1/2 * a * b * sin(C), where a and b are two sides of the triangle and C is the included angle.

Front

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

Back

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

Front

If a = 4, b = 5, and c = 8, how do you find angle B?

Back

Use the Law of Cosines: B = cos⁻¹((a² + c² - b²) / (2ac)). Substitute the values to find B.

Front

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

Back

In any triangle, the larger the angle, the longer the opposite side. This is a key principle in both the Law of Sines and the Law of Cosines.

Front

How do you solve for angles in a triangle using the Law of Cosines?

Back

Rearrange the Law of Cosines to isolate the cosine of the angle: cos(A) = (b² + c² - a²) / (2bc). Then use the inverse cosine function to find the angle.

Front

What is 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

When is the Law of Sines used instead of the Law of Cosines?

Back

The Law of Sines is used when you have two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA).

Front

What is the formula for the area of a triangle given two sides and the included angle?

Back

Area = 1/2 * a * b * sin(C), where a and b are the lengths of the two sides and C is the included angle.