Solving Triangles Using the Law of Cosines

Applications of Trigonometry

Solving Triangles: Law of Cosines

The Law of Cosines is a fundamental tool in trigonometry for solving triangles when certain combinations of sides and angles are known. It is especially useful when dealing with non-right triangles (oblique triangles) where the Law of Sines may not be applicable.

• Definition: The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles.

• Formula:

$c^2 = a^2 + b^2 - 2ab\cos(C)$

$b^2 = a^2 + c^2 - 2ac\cos(B)$

$a^2 = b^2 + c^2 - 2bc\cos(A)$

• When to Use: The Law of Cosines is used when you know:

  • Two sides and the included angle (SAS case)

  • All three sides (SSS case)

Example Problem

Given triangle ABC with sides $a = 7$, $b = 8$, and angle $C = 60^\circ$, find side $c$.

• Apply the Law of Cosines:

$c^2 = a^2 + b^2 - 2ab\cos(C)$

$c^2 = 7^2 + 8^2 - 2 \times 7 \times 8 \times \cos(60^\circ)$

$c^2 = 49 + 64 - 112 \times 0.5$

$c^2 = 113 - 56 = 57$

$c = \sqrt{57} \approx 7.55$

• Interpretation: The Law of Cosines allows us to solve for unknown sides or angles in non-right triangles.

Comparison: Law of Sines vs. Law of Cosines

Law of Sines	Law of Cosines
Used for ASA, AAS, or SSA cases	Used for SAS or SSS cases
$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$	$c^2 = a^2 + b^2 - 2ab\cos(C)$

• Note: In cases where you are given two sides and the included angle (SAS), the Law of Cosines is preferred because the Law of Sines would require knowing an angle opposite a known side.

Additional info: The handwritten notes and boxed explanations reinforce the correct application of the Law of Cosines and highlight the importance of choosing the appropriate law based on the given information in the triangle.