Maharashtra BoardEnglish MediumSTD 12 ScienceMathsTrigonometric Functions3 Marks
Question
Prove the Cosine rule using the Projection rule.
✓
Answer
Given: In $\triangle ABC , a = b \cos C + c \cos B$
$ b = c \cos A + a \cos C$
$c = a \cos B + b \cos A $
Mulitiply these equations by a,b,c respectively.
$ a^2=a b \cos C+a c \cos B$
$b^2=b c \cos A+a b \cos C$
$ c^2=a c \cos B+b c \cos A$
$ a^2+b^2-c^2=(a b \cos C+a c \cos B)+(b c \cos A+a b \cos C)-(a c \cos B+b c \cos A)$
$= a b \cos C+a c \cos B+b c \cos A+a b \cos C-a c \cos B-b c \cos A$
$=\quad & 2 a b \cos C$
$\therefore \quad & a^2+b^2-c^2=2 a b \cos C \quad \therefore c^2=a^2+b^2-2 a b \cos C .$
Similarly we can prove that
$a^2=b^2+c^2-2 b c \cos A \text { and } b^2=c^2+a^2-2 c a \cos B \text {. }$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.