Question

If $\angle A$ and $\angle B$ are acute angles such that $\cos A=\cos B$ , show that $\angle A=\angle B$.

Answer 1

Hint: In general opposite angles of equal sides are also equal. We can prove angle A equal to angle B by proving opposite sides of angle A and B also equal. Also cos of any angle equal to the ratio of adjacent to hypotenuse.

Complete step by step solution:
Let we have right angle triangle ABC as given below:

As we know cos is the ratio of adjacent to hypotenuse.
For angle B , adjacent = BC , hypotenuse = AB
So we can write
\[\cos B=\dfrac{BC}{AB}\] …………………………………(i)
For angle A , adjacent = AC , hypotenuse = AB
So we can write
\[\cos A=\dfrac{AC}{AB}\]…………………………………….(ii)
As given in question $\cos A=\cos B$.
So from equation (i) and (ii) we can write
$\dfrac{AC}{AB}=\dfrac{BC}{AB}$
$AC=BC$
According to the theorem, if two sides are equal then their opposite angles are also equal.
From triangle ABC , angle opposite to AC is B and angle opposite to BC is A.
So we can write
$\angle A=\angle B$
Hence proved.
Note: In the right angle triangle, angle is very important because we write adjacent and opposite according to angle.
So when we write cos for angle A, AC will be adjacent and for angle B, BC will be adjacent. We need to be careful about it.