Law of Sine and Cosine

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 for all three sides and angles in the triangle. It is 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 useful for finding a side when two sides and the included angle are known. It is expressed as: c² = a² + b² - 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

Find the length of side AC if AB = 10, BC = 12, and angle A = 60 degrees.

Back

AC = 14 (using the Law of Cosines).

Front

Find the distance a pilot travels if he must redirect from Norfolk to Raleigh, given the distances to Richmond and Raleigh are 143 miles and 160 miles respectively, and the angle between these paths is 45 degrees.

Back

Distance = 151 miles (using the Law of Cosines).

Front

What is the formula for the Law of Sines?

Back

The formula is: a/sin(A) = b/sin(B) = c/sin(C).

Front

What is the formula for the Law of Cosines?

Back

The formula is: c² = a² + b² - 2ab*cos(C).

Front

If angle A = 30 degrees and side a = 10, what is side b if angle B = 45 degrees?

Back

b = 10 * (sin(45)/sin(30)) = 14.14 (using the Law of Sines).

Front

Calculate the length of side x if angle A = 60 degrees, angle B = 45 degrees, and side a = 10.

Back

x = 10 * (sin(45)/sin(60)) = 8.66 (using the Law of Sines).