To find the value of sin 105 degrees, you can use the angle addition formula. The angle 105 degrees can be expressed as the sum of 60 degrees and 45 degrees:
sin(105°) = sin(60° + 45°)
Now, applying the sine addition formula, which states that sin(a + b) = sin(a)cos(b) + cos(a)sin(b), we can substitute:
sin(105°) = sin(60°)cos(45°) + cos(60°)sin(45°)
Next, we need to know the values of sin and cos for 60 degrees and 45 degrees:
- sin(60°) = √3/2
- cos(60°) = 1/2
- sin(45°) = √2/2
- cos(45°) = √2/2
Substituting these values into our equation gives:
sin(105°) = (√3/2)(√2/2) + (1/2)(√2/2)
This simplifies to:
sin(105°) = (√6/4) + (√2/4)
Combining these terms, we find:
sin(105°) = (√6 + √2)/4
Therefore, the value of sin 105 degrees is (√6 + √2)/4.