Exercise 34 Page 827

We are asked to derive the Law of Cosines by using the following triangle.

To do so, we will follow the given hints step by step.

• First, use the Pythagorean Theorem for △ DBC.
• In △ ADB, c^2= x^2+ h^2.
• cos A= x c.Now, let's start by applying the Pythagorean Theorem for △ DBC, which is the right-hand side triangle on the above figure.
  h^2+( b- x)^2= a^2

  ExpandNegPerfectSquare

  (a-b)^2=a^2-2ab+b^2

  h^2+b^2-2bx+x^2=a^2

  RearrangeEqn

  Rearrange equation

  a^2= h^2+ b^2-2 b x+ x^2

  Next, we will consider △ ADB, which is the left-hand side triangle, to apply the Pythagorean Theorem one more time and then rearrange it.

  c^2= x^2+ h^2

  SubEqn

  LHS-h^2=RHS-h^2

  c^2-h^2=x^2

  RearrangeEqn

  Rearrange equation

  x^2= c^2- h^2

  Great! Now we can substitute x^2= c^2- h^2 into the first equation that we found.

  a^2= h^2+ b^2-2 b x+ x^2

  Substitute

  x^2= c^()black2- h^()black2

  a^2=h^2+b^2-2bx+ c^2- h^2

  AddSubTerms

  Add and subtract terms

  a^2=b^2-2bx+c^2

  CommutativePropAdd

  Commutative Property of Addition

  a^2= b^2+ c^2-2 b x

  Let's now remember the last hint that is given to us. cos A=x/c ⇔ x= c * cos A Finally, we will substitute x= c * cos A into the equation a^2= b^2+ c^2-2 b x to get rid of the last x-variable.

  a^2= b^2+ c^2-2 b x

  Substitute

  x= c * cos A

  a^2=b^2+c^2-2b * ( c * cos A)

  Multiply

  Multiply

  a^2= b^2+ c^2-2 b c * cos A

  We ended with one of the Laws of Cosines, which we can use in cases when we know the measures of two sides and the included angle, or if we know the measures of three sides. Notice that we could also derive the other two formulas by rearranging the variables in different ways. a^2= b^2+ c^2-2 b c * cos A b^2= a^2+ c^2-2 a c * cos B c^2= a^2+ b^2-2 a c * cos C