Law of Sines and Cosines

Front

What is the Law of Sines?

Back

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

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).

Front

How do you determine the number of unique triangles that can be formed with given side lengths and angles?

Back

Use the Law of Sines or the Law of Cosines to analyze the given information. If the conditions lead to one valid triangle, it is unique; if two configurations are possible, there are two triangles; if no valid configuration exists, there are no triangles.

Front

What is the formula to find the area of a triangle using the Law of Sines?

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

If angle A = 30° and side a = 10, what is the length of side b if angle B = 45°?

Back

Using the Law of Sines: b = (a * sin(B)) / sin(A) = (10 * sin(45°)) / sin(30°) = 14.14.

Front

What is the ambiguous case in the Law of Sines?

Back

The ambiguous case occurs when two sides and a non-included angle (SSA) are known, leading to the possibility of zero, one, or two triangles.

Front

Given B = 70°, b = 85, and c = 88, how many triangles can be formed?

Back

2 Triangles can be formed.

Front

What is the significance of the angle opposite the longest side in a triangle?

Back

The angle opposite the longest side is always the largest angle in the triangle.

Front

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

Back

Rearrange the Law of Cosines formula: c = √(a² + b² - 2ab*cos(C)).

Front

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

Back

In a triangle, larger angles are opposite longer sides, and smaller angles are opposite shorter sides.