To find the value of sin 105 degrees, we can use the sine addition formula. First, we note that 105 degrees can be expressed as the sum of 60 degrees and 45 degrees:
105° = 60° + 45°
Now, we can apply the sine addition formula:
sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
In this case:
a = 60°, b = 45°
Let’s calculate the sine and cosine of the two angles:
- sin(60°) = √3/2
- cos(60°) = 1/2
- sin(45°) = √2/2
- cos(45°) = √2/2
Putting it all together:
sin(105°) = sin(60°)cos(45°) + cos(60°)sin(45°)
Substituting the values in:
sin(105°) = (√3/2)(√2/2) + (1/2)(√2/2)
This simplifies to:
sin(105°) = (√6/4) + (√2/4)
Combining the fractions:
sin(105°) = (√6 + √2) / 4
Thus, the value of sin 105 degrees is approximately 0.9659.