In trigonometry with angles measured in degrees, evaluate the exact value of the sum sin 225° + sin 45° using standard special angle values.

Introduction / Context:
This problem checks familiarity with sine values of standard angles, including angles in different quadrants. By combining sin 225° and sin 45°, the question encourages you to recall how the sine function behaves under angle transformations such as adding 180°. It is a straightforward but important skill used in many larger trigonometric simplifications and identities.

Given Data / Assumptions:

• Angles are given in degrees, not radians.
• We need to calculate sin 225° and sin 45°.
• We then add the two results: sin 225° + sin 45°.
• Standard exact values for special angles like 45° are assumed known.

Concept / Approach:
The main idea is to use reference angles and quadrant rules. The angle 45° lies in the first quadrant, where sine is positive. The angle 225° lies in the third quadrant and can be written as 180° + 45°. In the third quadrant, sine is negative, and the sine of 225° equals minus the sine of 45°. Recognising this relationship immediately suggests a cancellation when the two values are added.

Step-by-Step Solution:
1) Recall the standard value sin 45° = √2 / 2. 2) Express 225° as 180° + 45°. This shows that 225° is in the third quadrant. 3) In the third quadrant, sine is negative, and sin(180° + θ) = −sin θ. 4) Therefore sin 225° = sin(180° + 45°) = −sin 45°. 5) Substitute sin 45° = √2 / 2 to get sin 225° = −√2 / 2. 6) Now compute sin 225° + sin 45° = (−√2 / 2) + (√2 / 2). 7) The two terms cancel exactly, giving a total of 0.

Verification / Alternative check:
You can use a unit circle perspective to verify the signs and magnitudes. On the unit circle, the point corresponding to 45° has coordinates (√2 / 2, √2 / 2), so its sine value is √2 / 2. The point at 225° is directly opposite 45° on the circle, with coordinates (−√2 / 2, −√2 / 2), so its sine value is −√2 / 2. Adding these two y coordinates gives 0, which matches the algebraic reasoning. This geometric check confirms that there are no sign mistakes.

Why Other Options Are Wrong:
Option b (1) and option c (2) would require both sine values to be positive or to add up constructively, which they do not because sin 225° is negative. Option d (−1) would suggest that the negative contribution is larger in magnitude than the positive one, which is not the case here because their magnitudes are equal. Option e (4) is clearly far from the possible range of sine sums. Only option a (0) matches the exact cancellation that occurs.

Common Pitfalls:
A common error is to forget which quadrants make sine negative. Some learners also mis remember the value of sin 45°, confusing it with 1 / √2 and not simplifying consistently, even though these are equal. Another mistake is to mis write 225° as 180° − 45° instead of 180° + 45°, which would lead to incorrect quadrant reasoning. Carefully identifying the quadrant and using the reference angle removes these issues.

Final Answer:
Using standard trigonometric values and quadrant rules, the sum sin 225° + sin 45° simplifies to 0.