If ∠A and ∠B are acute angles such that cos A = cos B, then show that ∠A = ∠B.
Given: \(\angle A\) and \(\angle B\) are acute angles and \(\cos A = \cos B\).
To show: \(\angle A = \angle B\).
Step 1: Recall what “acute angle” means
An acute angle lies between \(0^\circ\) and \(90^\circ\). So,
\(0^\circ < A < 90^\circ\) and \(0^\circ < B < 90^\circ\).
Step 2: Use the key property of \(\cos\) in the acute range
On the interval \(0^\circ\) to \(90^\circ\), the cosine function is one-to-one (it gives a different value for each different acute angle). In simple words:
If two acute angles have the same cosine value, then the angles must be equal.
Step 3: Apply this property to the given statement
We are given \(\cos A = \cos B\).
Since both \(A\) and \(B\) are acute, cosine cannot take the same value for two different angles in this range.
Therefore, the only possibility is:
\(A = B\).
Conclusion: \(\angle A = \angle B\).
Student Note (why “acute” is important):
The condition “acute” matters because outside the acute range, cosine can repeat values. For example, \(\cos 60^\circ = \cos 300^\circ\), but \(60^\circ \ne 300^\circ\). In acute angles (\(0^\circ\) to \(90^\circ\)) this kind of repetition does not happen, so equality of cosines forces equality of angles.