Example  1
Use the cosine law to solve for the unknown side or angle. Round answers to the nearest tenth.



  1. To help you organize, write down what is given.

    \(\angle A = 20^\circ\)
    \(a = x\)
    \(b = 25 \thinspace \rm{cm}\)
    \(c = 22 \thinspace \rm{cm}\)

    \(\begin{align}
     a^2 &= b^2 + c^2 - 2bc\cos A \\
     x^2 &= \left( {25} \right)^2 + \left( {22} \right)^2 - 2\left( {25} \right)\left( {22} \right)\cos 20^\circ  \\
     x^2 &= 75.338... \\
     x &= 8.679... \\
     x &\doteq 8.7 \\
     \end{align}\)

    The unknown side length is approximately 8.7 cm.





  2. To help you organize, write down what is given.

    \(\angle A = \thinspace ?\)
    \(a = 61 \thinspace \rm{in}\)
    \(b = 42 \thinspace \rm{in}\)
    \(c = 89 \thinspace \rm{in}\)

    \(\begin{align}
     \cos A &= \frac{{b^2 + c^2 - a^2 }}{{2bc}} \\
     \cos A &= \frac{{\left( {42} \right)^2 + \left( {89} \right)^2 - \left( {61} \right)^2 }}{{2\left( {42} \right)\left( {89} \right)}} \\
     \cos A &= \frac{{5\thinspace 964}}{{7\thinspace 476}} \\
     \cos A &= 0.797... \\
     \angle A &= \cos ^{ - 1} \left( {0.797...} \right) \\
     \angle A &= 37.083...^\circ  \\
     \angle A &\doteq 37.1^\circ  \\
     \end{align}\)

    Note that you can use the regular version of the cosine law and solve for \(\angle A\).

    \(\begin{align}
     a^2 &= b^2 + c^2 - 2bc\cos A \\
     \left( {61} \right)^2 &= \left( {42} \right)^2 + \left( {89} \right)^2 - 2\left( {42} \right)\left( {89} \right)\cos A \\
     3\thinspace 721 &= 9\thinspace 685 - 7\thinspace 476\cos A \\
      - 5\thinspace 964 &=  - 7\thinspace 476\cos A \\
     0.797... &= \cos A \\
     \cos ^{ - 1} \left( {0.797...} \right) &= \angle A \\
     37.083...^\circ &= \angle A \\
     37.1^\circ &\doteq \angle A \\
     \end{align}\)
    The unknown angle is approximately \(37.1^\circ\).