Question

The value of \((\sin 30^\circ + \cos 30^\circ) - (\sin 60^\circ + \cos 60^\circ)\) is

• A. -1
• B. 0
• C. 1
• D. 2

Hint:

Recognize that \(\sin \theta = \cos(90^\circ - \theta)\) and \(\cos \theta = \sin(90^\circ - \theta)\). Here, \(\sin 30^\circ = \cos 60^\circ\) and \(\cos 30^\circ = \sin 60^\circ\). The expression is \((A+B)-(B+A)\), which is always zero.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This problem requires knowledge of the standard trigonometric ratios for specific angles, namely 30° and 60°.

Step 2: Key Formula or Approach:
We need the following standard values:
\(\sin 30^\circ = \frac{1}{2}\)
\(\cos 30^\circ = \frac{\sqrt{3}}{2}\)
\(\sin 60^\circ = \frac{\sqrt{3}}{2}\)
\(\cos 60^\circ = \frac{1}{2}\)

Step 3: Detailed Explanation:
Substitute these values into the given expression:
\[ (\sin 30^\circ + \cos 30^\circ) - (\sin 60^\circ + \cos 60^\circ) \] \[ = \left(\frac{1}{2} + \frac{\sqrt{3}}{2}\right) - \left(\frac{\sqrt{3}}{2} + \frac{1}{2}\right) \] Since the terms inside both parentheses are identical, subtracting them will result in zero.
\[ = \frac{1 + \sqrt{3}}{2} - \frac{\sqrt{3} + 1}{2} \] \[ = 0 \]

Step 4: Final Answer:
The value of the expression is 0. This matches option (B).