Law of Sine and Law of Cosine

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 expressed as: \( c^2 = a^2 + b^2 - 2ab \cos(C) \)

Front

When do you use the Law of Sines?

Back

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

Front

When do you use the Law of Cosines?

Back

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

Front

Calculate the length of side c in a triangle with sides a = 5, b = 7, and angle C = 60°.

Back

Using the Law of Cosines: \( c^2 = 5^2 + 7^2 - 2 \cdot 5 \cdot 7 \cdot \cos(60°) \) \( c^2 = 25 + 49 - 35 = 39 \) \( c = \sqrt{39} \approx 6.24 \)

Front

Find the area of a triangle with sides a = 6 m, b = 8 m, and included angle C = 137°.

Back

Area = \( \frac{1}{2}ab \sin(C) = \frac{1}{2} \cdot 6 \cdot 8 \cdot \sin(137°) \approx 16.4 m² \)

Front

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

Back

Area = \( \frac{1}{2}ab \sin(C) \), where a and b are the lengths of two sides and C is the included angle.

Front

If angle A = 30° and angle B = 45°, what is angle C in a triangle?

Back

Angle C can be found using the triangle sum property: \( C = 180° - A - B = 180° - 30° - 45° = 105° \)

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. This is a key principle in both the Law of Sines and the Law of Cosines.

Front

How do you find an unknown angle using the Law of Sines?

Back

To find an unknown angle A, rearrange the Law of Sines: \( \sin(A) = \frac{a \cdot \sin(B)}{b} \) and then use the inverse sine function.