Question 5 of 18intermediate🔍 AnalyzeShort Answer2 marks

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

Correct Answer

Given that ∠A and ∠B are acute angles, i.e., 0° < A, B < 90°. From Table 8.1, we observe that cos A decreases from 1 to 0 as A increases from 0° to 90°. Specifically, cos 0° = 1, cos 30° = 3\sqrt{3}/2, cos 45° = 1/2\sqrt{2}, cos 60° = 1/2, cos 90° = 0. This shows that cosine is a strictly decreasing function for acute angles. Therefore, if cos A = cos B, it necessarily implies that A = B. Hence, ∠A = ∠B.

Exercise: EXERCISE 8.1 | Q: 6 | (Page Page 9)
For More Understanding

Explanation

The textbook provides a table of trigonometric values and a remark that cos A decreases as A increases from 0° to 90°. This property ensures that cosine is one-to-one on the interval of acute angles, so equality of cosines forces equality of angles. This is a direct application of the observation from the context.

Solution Steps

  1. Step 1: Note that A and B are acute, so 0° < A, B < 90°.

  2. Step 2: Refer to Table 8.1 which shows cos values: cos 0°=1, cos 30°=3\sqrt{3}/2, cos 45°=1/2\sqrt{2}, cos 60°=1/2, cos 90°=0.

  3. Step 3: Observe that as angle increases, cos value decreases; hence cos is strictly decreasing on (0°,90°).

  4. Step 4: Since cos is one-to-one, cos A = cos B implies A = B.

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