To find the value of sin 105Β°, we can use the formula sin(π΄ + π΅) = sin π΄ cosπ΅ + cos π΄ sinπ΅.
Let's express 105Β° as a sum of two angles: 105Β° = 60Β° + 45Β°.
Using the formula, we have:
sin(105Β°) = sin(60Β° + 45Β°)
= sin(60Β°)cos(45Β°) + cos(60Β°)sin(45Β°)
Now, we know that sin 60Β° = β3/2, cos 60Β° = 1/2, sin 45Β° = β2/2, and cos 45Β° = β2/2.
Substituting these values into the formula, we get:
sin(105Β°) = (β3/2)(β2/2) + (1/2)(β2/2)
= (β3β2)/4 + β2/4
= (β6 + β2)/4
= (β6 + β2)/2β2
To rationalize the denominator, we multiply the numerator and denominator by β2:
= [(β6 + β2)/2β2] * (β2/β2)
= (β12 + 2)/4β2
= (β12 + 2)/4β2 * (β2/β2)
= (β24 + 4β2)/8
= (β24 + 4β2)/8
We notice that β24 = β(4 * 6) = 2β6.
Therefore, we have:
sin(105Β°) = (2β6 + 4β2)/8
= β6/4 + β2/2
= β6/4 + (2/2)β2
= β6/4 + β2/1
= (β6 + 4β2)/4
= (1/4)(β6 + 4β2)
Comparing this result with the given options, we find that the answer is:
A. β42 (β3 + 1)