Chapter 9.6, Problem 7E

Solution

To decide: that to solve the triangle with the given case whether the law of sines or the law of cosines should be used.

Law of sines

Given information:

The given case is:

AAS

Definition Used:

Law of Cosines:

If $△ABC$

has sides of length a , b , and c as shown, then:

  

  $\begin{array}{l}{a}^2=b^2+c^2-2bc\cos A\\ b^2=a^2+c^2-2ac\cos B\\ c^2=a^2+b^2-2ab\cos C\end{array}$

Law of Sines:

The law of sines can be written in either of the following forms for $△ABC$ with sides of length a , b and c .

  

  $\begin{array}{l}\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}\\ \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\end{array}$

Explanation:

Since, two of the angle measures of the triangle are given. It is easy to find the measure of the third angle of the triangle using the fact that the sum of the measures of all angles of a triangle is $180°$ .

And as one side length of the triangle is given, so it is easy to find the other two side lengths using the law of sines.

However, if one wish to use law of cosines measure of at least two side lengths and the included angle must be known which is not the given case. (So even if one wants to use law of cosine, first it will be required to use law of sines to find measure of another side length.)

Therefore, in this case, the law of sines should be used.