6.5 The Law of Cosines

6.5.1 Learning Objectives

By the end of this section, you should be able to:

  • apply the Law of Cosines;
  • solve SAS and SSS triangles;
  • choose between the Laws of Sines and Cosines;
  • calculate triangle area using Heron’s formula; and
  • solve applied distance and bearing problems.

6.5.2 The Law of Cosines

For triangle \(ABC\),

\[ c^2=a^2+b^2-2ab\cos C, \]

with corresponding forms for \(a^2\) and \(b^2\). Use it for SAS or SSS information. When \(C=90^\circ\), it becomes the Pythagorean Theorem.

Example 6.14 Solving an SAS Triangle

Given \(a=9\), \(b=13\), and \(C=58^\circ\), find \(c\).

Solution

\[ c=\sqrt{9^2+13^2-2(9)(13)\cos58^\circ}\approx11.22. \]

\[ \boxed{c\approx11.22} \]

Exercise 6.14 Given \(a=7\), \(b=10\), and \(C=60^\circ\), find \(c\).

Show answer

Answer: \(c^2=49+100-70=79\), so \(\boxed{c=\sqrt{79}\approx8.89}\).

6.5.3 Solving SSS Triangles

When all three sides are known, isolate a cosine:

\[ \cos C=\frac{a^2+b^2-c^2}{2ab}. \]

Find the largest angle first, opposite the longest side.

Example 6.15 Finding an Angle from Three Sides

Find the angle opposite side \(12\) in a triangle with sides \(7\), \(10\), and \(12\).

Solution

\[ \cos C=\frac{7^2+10^2-12^2}{2(7)(10)}=\frac5{140}. \]

\[ \boxed{C=\cos^{-1}(1/28)\approx87.95^\circ} \]

Exercise 6.15 Find the largest angle in a triangle with sides \(5\), \(6\), and \(8\).

Show answer

Answer: \(\boxed{\cos^{-1}((5^2+6^2-8^2)/(2\cdot5\cdot6))\approx92.87^\circ}\)

6.5.4 Area and Heron’s Formula

With two sides and their included angle,

\[ K=\frac12ab\sin C. \]

With three sides, define \(s=(a+b+c)/2\). Heron’s formula is

\[ K=\sqrt{s(s-a)(s-b)(s-c)}. \]

Example 6.16 Finding Area from Three Sides

Find the area of a triangle with sides \(7\), \(8\), and \(9\).

Solution

\[ s=\frac{7+8+9}{2}=12. \]

\[ K=\sqrt{12(5)(4)(3)}=\sqrt{720}=12\sqrt5. \]

\[ \boxed{K=12\sqrt5\approx26.83\text{ square units}} \]

Exercise 6.16 Find the area when \(a=10\), \(b=14\), and \(C=40^\circ\).

Show answer

Answer: \(\boxed{K=\frac12(10)(14)\sin40^\circ\approx44.99\text{ square units}}\)

6.5.5 Conceptual Takeaways

  • The Law of Cosines extends the Pythagorean Theorem.
  • SAS and SSS information point to the Law of Cosines.
  • The longest side is opposite the largest angle.
  • Heron’s formula finds area from three sides.
  • A complete solution includes a labelled diagram, units, and sensible rounding.

6.5.6 Skills You Should Be Able to Do

  • Choose the correct Law of Cosines form.
  • Solve SAS and SSS triangles.
  • Select between sine and cosine methods.
  • Calculate area using two sides and an angle or Heron’s formula.
  • Apply oblique-triangle methods to measurement.

6.5.7 Practice Problems with Solutions

  1. Given \(a=6\), \(b=11\), and \(C=45^\circ\), find \(c\).

    Show Solution

    \[ c=\sqrt{6^2+11^2-2(6)(11)\cos45^\circ}\approx\boxed{7.98}. \]

  2. Given \(a=9\), \(b=12\), and \(c=15\), find \(C\).

    Show Solution

    \[ \cos C=\frac{9^2+12^2-15^2}{2(9)(12)}=0. \]

    Thus, \(\boxed{C=90^\circ}\).

  3. Find the largest angle in a triangle with sides \(8\), \(13\), and \(15\).

    Show Solution

    The largest angle \(C\) is opposite \(15\):

    \[ \cos C=\frac{8^2+13^2-15^2}{2(8)(13)}=\frac8{208}. \]

    \[ \boxed{C\approx87.80^\circ} \]

  4. Find the area of a triangle with sides \(5\), \(5\), and \(6\).

    Show Solution

    \[ s=8,\qquad K=\sqrt{8(3)(3)(2)}=\sqrt{144}. \]

    \[ \boxed{K=12\text{ square units}} \]

  5. Find the area when \(a=18\), \(b=24\), and \(C=35^\circ\).

    Show Solution

    \[ K=\frac12(18)(24)\sin35^\circ\approx\boxed{123.89\text{ square units}}. \]

  6. Two paths of lengths \(140\text{ m}\) and \(190\text{ m}\) leave the same point with an included angle of \(72^\circ\). Find the distance between their endpoints.

    Show Solution

    \[ d=\sqrt{140^2+190^2-2(140)(190)\cos72^\circ}\approx\boxed{198.14\text{ m}}. \]