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: Evaluate Assertion (A).
Assertion A states: For an acute angle \(\theta\), \(\cos\theta\) is always less than 1. For strictly acute angles \(0^\circ < \theta < 90^\circ\), \(\cos\theta\) ranges from just below 1 to just above 0, always less than 1. Assertion A is TRUE.
Step 2: Evaluate Reason (R).
Reason R states: In a right-angled triangle, the hypotenuse is the longest side and \(\cos\theta = \frac{\text{Base}}{\text{Hypotenuse}}\). This is correct - hypotenuse is always the longest side, making Base less than Hypotenuse, hence \(\cos\theta < 1\). Reason R is TRUE.
Step 3: Check if R correctly explains A.
Because Base < Hypotenuse in a right triangle, \(\cos\theta = \frac{\text{Base}}{\text{Hypotenuse}} < 1\). So R directly justifies why A is true.
Step 4: Eliminate other options.
Option 2 requires R to not explain A (but it does). Options 3 and 4 require one statement to be false (both are true).
Step 5: Confirm the relationship.
R is the logical and complete reason for A. Both A and R are true, and R correctly explains A.
Step 6: Select the correct option.
Option 1 is correct.
\[ \boxed{\text{Both A and R are true and R is the correct explanation of A.}} \]