Question

If Angle A and angle B are acute angles such that cos A = Cos B,then show that angle a = angle b​

Answer 1

Answer

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sreeranjinikkumar

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Step-by-step explanation:

In triangle ABC ,

Cos A = opposite side / hypo.

= AC / AB -1

Cos B = adjacent side / hypo.

= BC / AB -2

We gave cos A = cos B

Thus by 1 and 2,

AC / AB = BC / AB

Thus

AC . AB = BC . AB

AC = BC

thus AC and BC are two sides of triangle ABC

Thus angle A = angle B

( angles opposite to equal sides are equal )

This is the answer .

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Answer 2

We are given that,

$\qquad \rm {• cos \: A = cos \: B }$

________

We have to show,

$\qquad \rm {• ∠A = ∠B }$

________

Construction,

$\dag$ Draw a triangle ABC where C = 90°.

__________

Solution

We know,

$\qquad \rm {• cos \: A = cos \: B }$

$\implies { \rm \dfrac{B}{H} = \dfrac{B}{H} }$

$\implies { \rm \dfrac{AC}{AB} = \dfrac{BC}{AB} }$

$\implies \rm{ AC \times \cancel {AB} = BC \times \cancel {AB} }$

$\rm {\implies AC = BC }$

Now , we know that if two sides of triangle are equal than it is isosceles triangle and we also know that in isosceles triangle, if two opposite sides are equal than their angle will also be equal.

$\implies { \rm \angle A= \angle B }$

Thus, prooved!