Evaluate ($\sin$45º + $\sin$30º ) in surd form

Analyze the question: We need the exact (surd) values of sin 45° and sin 30°, then add them.

Concept: These are standard exact values you must know:

• sin 30° = 1/2
• sin 45° = 1/√2 = √2/2

Solution:

sin 45° + sin 30° = √2/2 + 1/2

Put them over a common denominator:

= (√2 + 1)/2 = (1 + √2)/2

Check the options:

• A. √3/(2√2) ≈ 0.612. No.
• B. √3 − 1/2 ≈ 1.232. No.
• C. (1/2)√2 ≈ 0.707. This is only sin 45°, so it leaves out sin 30°.
• D. (1 + √2)/2 ≈ 1.207. Matches our result (0.7071 + 0.5 = 1.2071).

Tip: Memorize the sine table for 30°, 45° and 60°: 1/2, √2/2, √3/2. The values go up in the pattern √1/2, √2/2, √3/2. Cosine is the same list in reverse. When you see "surd form," rationalize any 1/√2 into √2/2 and combine over a common denominator.