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Grade: 12

                        

If ∠A and ∠B are acute angles such that cos A = cos B, then show that ∠ A = ∠ B. plixxzxxxx

one month ago

Answers : (1)

Anand Kumar Pandey
askIITians Faculty
1372 Points
							Dear Student

Let us assume the triangle ABC in which CD⊥AB
Given that the angles A and B are acute angles,
such that cos (A) = cos (B)
As per the angles taken,
the cos ratio is written as AD/AC = BD/BC
Now, interchange the terms, we get
AD/BD = AC/BC
Let take a constant value
AD/BD = AC/BC = k

Now consider the equation as

AD = k BD ...(1)

AC = kBC ...(2)

By applying Pythagoras theorem in△CAD and△CBD we get,

CD^2 = BC^2 –BD^2... (3)
CD^2 =AC^2−AD^2 ....(4)

From the equations (3) and (4) we get,
AC^^2−AD2= BC^2−BD^2
Now substitute the equations (1) and (2) in (3) and (4)
K^2(BC^2−BD^2)=(BC^2−BD^2) k^2=1
Putting this value in equation, we obtain
AC = BC
∠A=∠B (Angles opposite to equal side are equal-isosceles triangle)

Thanks
one month ago
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