Final answer:
The exact value of sin(105 degrees) is found using the sine addition formula. By rewriting 105 degrees as 60 degrees plus 45 degrees, we get sin(105) = (sqrt(2) + sqrt(3))/2, which is option d.
Step-by-step explanation:
To find the exact value of sin(105 degrees), we can use the sine addition formula, sin(a + b) = sin(a)cos(b) + cos(a)sin(b). Now, 105 degrees can be rewritten as the sum of 60 degrees and 45 degrees, angles for which we know the exact values of sine and cosine. Thus:
sin(105°) = sin(60° + 45°)
= sin(60°)cos(45°) + cos(60°)sin(45°)
= (√3/2)(√2/2) + (1/2)(√2/2)
= √6/4 + √2/4
= (√6 + √2)/4
As a fraction of the square roots, we multiply the numerator and the denominator by 2 to get rid of the fraction within the fraction:
(√6 + √2)/4 * 2/2 = (2√6 + 2√2)/8 = (√6 + √2)/4
Therefore, the exact value of sin(105 degrees) is (√2 + √3)/2, which corresponds to option d.