Question

if angle A and angleB are acute angles such that cos A =COS B,show that angle A = angle B

Answer 1

Answer:

According to the question:-

Let cosA=

Hypotenuse

sideadjacentA

AB

AC

Similarly,

cosB=

Hypotenuse

sideadjacentB

=

AB

BC

Given that

cosA=cosB

AB

AC

=

AB

BC

AC=BC

In triangle,

angles opposite equal side are equal

∠B=∠A

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!