HSC Arts 12th Board ExamMaharashtra State Board
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In any ΔABC, with usual notations, prove that b2=c2+a2−2cacosB - HSC Arts 12th Board Exam - Mathematics and Statistics

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Question

In any ΔABC, with usual notations, prove that `b^2=c^2+a^2-2ca cosB`.

Solution

Consider that for ΔABC,∠B is in a standard position i.e. vertex B is at the origin and the side BC is along positive x-axis. As ∠B is an angle of a triangle ∠B can be acute or B ∠B can be obtuse.

 

Using the Cartesian co-ordinate system in both figure (1) and figure (2)
we get B ≡ (0,0)A ≡ (c cos B, c sin B) and C ≡ (a,0)

Now consider l(CA) =b

`therefore b^2=(a-ccosB)^2+(0-csinB)^2` , by distance formula

`b^2=a^2-2ac cosB+c^2cos^2B+c^2sin^2B`

`b^2=a^2-2ac cosB+c^2(sin^2B+cos^2B)`

`b^2=a^2+c^2-2ac cosB`

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APPEARS IN

 2013-2014 (March) (with solutions)
Question 2.2.1 | 4.00 marks
Solution In any ΔABC, with usual notations, prove that b2=c2+a2−2cacosB Concept: Trigonometric Functions - Solution of a Triangle.
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