Q-In-AB-C-cos-A-cos-B-2cos-C-2-show-that-a-b-2c-

Question Number 212552 by mnjuly1970 last updated on 17/Oct/24

\overset{Q :}{} In A \overset{Δ}{B C} : c o s (A) + c o s (B) + 2 c o s (C) = 2

show that : a + b = 2 c

Answered by Ghisom last updated on 17/Oct/24

C = π - (A + B) \Rightarrow

\cos A + \cos B - \cos (A + B) = 2

c = \frac{a + b}{2} \Rightarrow

a^{2} + b^{2} + 2 a b \cos C = c^{2} etc . lead to

(1) \cos A = \frac{5 b - 3 a}{4 b} \Rightarrow

\sin A = \frac{\sqrt{- 3 (a - 3 b) (3 a - b)}}{4 b}

(2) \cos B = \frac{5 a - 3 b}{4 a} \Rightarrow

\sin B = \frac{\sqrt{- 3 (a - 3 b) (3 a - b)}}{4 a}

(3) \cos A \cos B - \sin A \sin B = - \frac{3 a^{2} - 2 a b + 3 b^{2}}{8 a b}

\cos (A + B) = \cos A \cos B - \sin A \sin B

inserting from above

\Rightarrow true

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