H-Law of Sines/Cosines and Area Flashcard

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: \( \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \)

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 given by: \( c^2 = a^2 + b^2 - 2ab \cos(C) \)

Front

How do you find the area of a triangle using the Law of Sines?

Back

The area \( A \) of a triangle can be calculated using the formula: \( A = \frac{1}{2}ab \sin(C) \), where \( a \) and \( b \) are the lengths of two sides and \( C \) is the included angle.

Front

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

Back

The area \( A \) can also be calculated using the formula: \( A = \frac{1}{2}ab \sin(C) \), which is derived from the Law of Sines.

Front

What is the significance of the angle of elevation in triangle problems?

Back

The angle of elevation is the angle formed by the horizontal line and the line of sight to an object above the horizontal. It is crucial for solving problems involving heights and distances.

Front

How do you determine the number of possible triangles given certain measurements?

Back

To determine the number of possible triangles, use the given angles and sides to check for the existence of a triangle using the Triangle Inequality Theorem and the Law of Sines.

Front

What is the Triangle Inequality Theorem?

Back

The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Front

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

Back

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

Front

How do you solve for a missing side using the Law of Sines?

Back

To solve for a missing side using the Law of Sines, rearrange the formula: \( a = \frac{b \sin(A)}{\sin(B)} \) where \( A \) and \( B \) are known angles and \( b \) is the known side.

Front

What is the purpose of using the Law of Cosines?

Back

The Law of Cosines is used to find a side of a triangle when two sides and the included angle are known, or to find an angle when all three sides are known.