Question

If \( a = \sin 175^\circ + \cos 175^\circ \), then

• A. \( a>0 \)
• B. \( a = 0 \)
• C. \( a<0 \)
• D. \( a = 1 \)

Hint:

When adding sine and cosine functions, consider their signs and relative values for the given angle.

Answer 1

The Correct Option is C

Step 1: Evaluate the trigonometric values.
We are given that \( a = \sin 175^\circ + \cos 175^\circ \). Using the angle values: \[ \sin 175^\circ = \sin (180^\circ - 5^\circ) = \sin 5^\circ, \quad \cos 175^\circ = -\cos 5^\circ. \] Thus, \[ a = \sin 5^\circ - \cos 5^\circ. \] Since \( \sin 5^\circ \) is positive and \( \cos 5^\circ \) is also positive but smaller than \( \sin 5^\circ \), we have \( a<0 \).
Step 2: Conclusion.
Thus, the correct answer is \( a<0 \), corresponding to option (C).