Q .5. - Calculus

Question

Why is the Law of Cosines a generalization of the Pythagorean theorem?

Step-by-Step Solution

Answer

As the law of cosines is $a^2+b^2=c^2$

Step 1:Given information

 the Law of Cosines  

Step 2:Explaination

 In a triangle with sides $a, b$  and  c , where$\theta$  is the angle between the sides of length $a$ and $b$ then

$a^2+b^2-2ab\cos 90°=c^2.$

$\text{ Using }\cos 90°=0\text{, the expression }{a}^2+b^2-2ab\cos 90°=c^2\text{ reduces to }$

$a^2+b^2-2ab\left(0\right)=c^2$

hence

$a^2+b^2-2ab\cos {90}^{\text{'}\text{'}}=c^2$

$a^2+b^2=c^2$

Which is Pythagoras theorem.

Hence, the law of cosines is the generalization of Pythagoras theorem.