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

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 Heron's Formula for the area of a triangle?

Back

Heron's Formula states that the area \( A \) of a triangle with sides of lengths \( a, b, c \) is given by: \( A = \sqrt{s(s-a)(s-b)(s-c)} \), where \( s = \frac{a+b+c}{2} \) is the semi-perimeter.

Front

How do you determine if a triangle can be formed with given sides?

Back

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

Front

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

Back

The angle of elevation is the angle between 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 apply the Law of Sines to find unknown sides?

Back

To find an unknown 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 relationship between angles and sides in the Law of Sines?

Back

In the Law of Sines, larger angles correspond to longer opposite sides, and smaller angles correspond to shorter opposite sides.

Front

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

Back

To find an angle using the Law of Cosines, rearrange the formula: \( C = \cos^{-1}\left(\frac{a^2 + b^2 - c^2}{2ab}\right) \) where \( a, b, c \) are the lengths of the sides.

Front

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

Back

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