Chapter 8.3, Problem 63AYU

To determine

State the Law of Cosines in words.

Answer to Problem 63AYU

  $\boxed{a^2=b^2+c^2-2b\cos \alpha }$ , $\boxed{b^2=a^2+c^2-2ac\cos \beta }$

Explanation of Solution

Given information:

State the Law of Cosines in words.

Calculation:

Law of Cosines relates the lengths of the sides of the plane triangle to the cosine of any of its angle. It is applied in triangles either having its all three sides or two sides and its included angle. Thus it is useful for computing the third side of a triangle if two sides and included angle is given and in computing the angles of triangle if all three sides are given.

According to this Law,

“For a triangle with sides a,b, and c and the angle $\theta$ opposite the side c, we have

  $c^2=a^2+b^2-2ab\cos \theta$ ”

The formula above can also be represented in other form:

  $\cos C=\frac{a^2+b^2-c^2}{2ab}$

Hence, By changing which sides of the triangle play the roles of a,b, and c in the original formula ,one discovers that the following two formulas also state the law of cosines:

  $a^2=b^2+c^2-2b\cos \alpha$

  $b^2=a^2+c^2-2ac\cos \beta$