Law of Sines / Law of 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).

Tags

CCSS.HSG.SRT.D.10

CCSS.HSG.SRT.D.11

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).

Tags

CCSS.HSG.SRT.D.10

CCSS.HSG.SRT.D.11

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).

Tags

CCSS.HSG.SRT.D.10

CCSS.HSG.SRT.D.11

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.

Tags

CCSS.HSG.SRT.D.10

CCSS.HSG.SRT.D.11

Front

Calculate the length of side a if angle A = 30°, angle B = 45°, and side b = 10.

Back

a = b * (sin(A) / sin(B)) = 10 * (sin(30°) / sin(45°)) = 10 * (0.5 / 0.7071) ≈ 7.07.

Tags

CCSS.HSG.SRT.D.10

CCSS.HSG.SRT.D.11

Front

If side a = 8, side b = 6, and angle C = 60°, find side c using the Law of Cosines.

Back

c² = a² + b² - 2ab*cos(C) = 8² + 6² - 2*8*6*cos(60°) = 64 + 36 - 48 = 52. Thus, c ≈ 7.21.

Tags

CCSS.HSG.SRT.D.10

CCSS.HSG.SRT.D.11

Front

What is the sine of 90 degrees?

Back

The sine of 90 degrees is 1.

Tags

CCSS.HSG.SRT.C.7

Front

What is the cosine of 0 degrees?

Back

The cosine of 0 degrees is 1.

Tags

CCSS.HSG.SRT.C.7

Front

If angle A = 30° and angle B = 60°, what is angle C?

Back

Angle C = 180° - (A + B) = 180° - (30° + 60°) = 90°.

Tags

CCSS.7.G.B.5

Front

In triangle ABC, if angle A = 45° and side a = 10, what is the relationship between sides a, b, and c using the Law of Sines?

Back

a/sin(A) = b/sin(B) = c/sin(C).

Tags

CCSS.HSG.SRT.D.10

CCSS.HSG.SRT.D.11