Law of Cosines Notes

Law of Cosines

Introduction

The law of cosines is used to solve triangles when the law of sines cannot be applied, specifically when there is not enough information to set up proportions with one unknown.

When to use Law of Cosines

  • When you can't use the Law of Sines. Primarily when you don't have an angle and its opposite side.
  • Side-Angle-Side (SAS) triangles.
  • Side-Side-Side (SSS) triangles.

Law of Cosines Formula

The law of cosines is a generalization of the Pythagorean theorem that applies to all triangles, not just right triangles. The standard form of the law of cosines is:

c2=a2+b2−2ab⋅cos(C)c^2 = a^2 + b^2 - 2ab \cdot cos(C)

Where:

  • cc is the side opposite angle CC.
  • aa and bb are the other two sides of the triangle.
  • CC is the angle opposite side cc.

The Law of Cosines can be rearranged to solve for any of the missing angles or sides of a triangle.

Finding a Side

Given two sides and the included angle (the angle between them), you can find the length of the third side using the law of cosines.

Example:

Given:

  • a=10a = 10
  • b=8b = 8
  • C=110∘C = 110^\circ

Find cc.

Applying the law of cosines:

c2=102+82−2⋅10⋅8⋅cos(110∘)c^2 = 10^2 + 8^2 - 2 \cdot 10 \cdot 8 \cdot cos(110^\circ)

c2=100+64−160⋅cos(110∘)c^2 = 100 + 64 - 160 \cdot cos(110^\circ)

Calculating the value:

c2≈164−160⋅(−0.342)c^2 ≈ 164 - 160 \cdot (-0.342)

c2≈164+54.72c^2 ≈ 164 + 54.72

c2≈218.72c^2 ≈ 218.72

c≈218.72c ≈ \sqrt{218.72}

c≈14.79c ≈ 14.79

Finding an Angle

Given three sides of a triangle, you can find any of the angles using the law of cosines.

Example Usage:

Given:

  • a=5a = 5
  • b=4b = 4
  • c=8c = 8

Find angle CC.

Start with the law of cosines:

c2=a2+b2−2ab⋅cos(C)c^2 = a^2 + b^2 - 2ab \cdot cos(C)

Rearrange to solve for cos(C)cos(C):

cos(C)=c2−a2−b2−2abcos(C) = \frac{c^2 - a^2 - b^2}{-2ab}

cos(C)=82−52−42−2⋅5⋅4cos(C) = \frac{8^2 - 5^2 - 4^2}{-2 \cdot 5 \cdot 4}

cos(C)=64−25−16−40cos(C) = \frac{64 - 25 - 16}{-40}

cos(C)=23−40cos(C) = \frac{23}{-40}

cos(C)=−0.575cos(C) = -0.575

Find angle CC by taking the inverse cosine:

C=cos−1(−0.575)C = cos^{-1}(-0.575)

C≈125.1∘C ≈ 125.1\circ

Notes

  • The law of cosines avoids the ambiguous case that can occur with the law of sines.
  • When using the law of cosines, ensure your calculator is in the correct mode (degrees or radians).
  • The angle found using the inverse cosine function will always be between 0∘0^\circ and 180∘180^\circ.