If ∠B and ∠Q are acute angles such that sinB=sinQ, then prove that ∠B=∠Q.
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Step-by-Step Solution
Step 1: Define acute angles and sine function
Acute angles are angles whose measures are greater than 0∘ and less than 90∘. The sine function relates an angle of a right-angled triangle to the ratio of the length of the opposite side to the length of the hypotenuse.
Step 2: Consider the given condition
We are given that the sine of angle B is equal to the sine of angle Q.
Step 3: Analyze the sine function for acute angles
The sine function is strictly increasing in the interval [0∘,90∘]. This means that for any two distinct acute angles, their sines will be distinct. Therefore, if the sines of two acute angles are equal, the angles themselves must be equal.
Step 4: Conclude the proof
Since both ∠B and ∠Q are acute angles, and their sines are equal (sinB=sinQ), it logically follows that the angles themselves must be equal.