Question

Assertion (A) : For an acute angle \(\theta\), \(\cos \theta\) is always less than 1.
Reason (R) : In a right-angled triangle, hypotenuse is the longest side and \(\cos \theta = \frac{\text{Base}}{\text{Hypotenuse}}\).

• A. Both, Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
• B. Both, Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
• C. Assertion (A) is true, but Reason (R) is false.
• D. Assertion (A) is false, but Reason (R) is true.

Hint:

To test if Reason (R) is the correct explanation for Assertion (A), read them together using "because":
"For an acute angle \(\theta\), \(\cos \theta\) is always less than 1 because \(\cos \theta = \frac{\text{Base}}{\text{Hypotenuse}}\) and the hypotenuse is always the longest side."
Since this statement is mathematically sound and logical, the answer is (A).

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
This is an Assertion-Reason style question based on the fundamentals of Trigonometry.
We need to evaluate whether the Assertion (A) and the Reason (R) are individually true, and if so, whether the Reason correctly explains the Assertion.

Step 2: Key Formula or Approach:
1. An acute angle is an angle \(\theta\) such that \(0^\circ \lt \theta \lt 90^\circ\).
2. In a right-angled triangle, the cosine of an angle \(\theta\) is defined as: \[ \cos \theta = \frac{\text{Base}}{\text{Hypotenuse}} \] 3. By the properties of a right-angled triangle, the side opposite to the right angle (the hypotenuse) is always strictly longer than both the base and the perpendicular.

Step 3: Detailed Explanation:
1. Analyze the Assertion (A):
- For any acute angle \(\theta\) (where \(0^\circ \lt \theta \lt 90^\circ\)), the value of \(\cos \theta\) lies strictly between 0 and 1.
- Specifically, \(\cos 0^\circ = 1\) and \(\cos 90^\circ = 0\), so for any angle strictly between them, \(\cos \theta \lt 1\).
- Thus, the Assertion (A) is true.
2. Analyze the Reason (R):
- In any right-angled triangle, the Hypotenuse is the longest side. This is true because the right angle (\(90^\circ\)) is the largest angle, and the side opposite the largest angle is always the longest.
- The definition of cosine is indeed \(\cos \theta = \frac{\text{Base}}{\text{Hypotenuse}}\).
- Thus, the Reason (R) is also true.
3. Check the Connection (Explanation):
- Since \(\text{Hypotenuse} \gt \text{Base}\), dividing a smaller positive number (Base) by a larger positive number (Hypotenuse) must result in a value strictly less than 1.
- Mathematically: \[ \text{Base} \lt \text{Hypotenuse} \implies \frac{\text{Base}}{\text{Hypotenuse}} \lt 1 \implies \cos \theta \lt 1 \] - Therefore, the Reason correctly explains why \(\cos \theta\) is always less than 1 for an acute angle.

Step 4: Final Answer:
Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).
Therefore, the correct option is (A).