To find the value of sin 15 degrees, we can utilize the sine subtraction formula, which states:
sin(a – b) = sin(a)cos(b) – cos(a)sin(b)
For our case, we can set:
- a = 30 degrees
- b = 15 degrees
This means:
sin(15) = sin(30 – 15)
Substituting into the formula:
sin(15) = sin(30)cos(15) – cos(30)sin(15)
We know the following values:
- sin(30) = 0.5
- cos(30) = √3/2
- cos(15) = √(1 – sin²(15))
To express sin(15) in terms of known values, we rearrange our equation:
sin(15) = 0.5 * cos(15) – (√3/2) * sin(15)
Now, we can isolate sin(15) on one side:
sin(15) + (√3/2) * sin(15) = 0.5 * cos(15)
Factoring out sin(15):
sin(15) * (1 + √3/2) = 0.5 * cos(15)
Now, we need the value of cos(15), which can be computed, but since we are simplifying our calculations, let’s directly calculate:
We can use the known identity:
sin(15) = √6/4 – √2/4
Thus, we find:
sin(15) = (√6 – √2) / 4
In summary, the value of sin 15 degrees, derived using sin 30 degrees through the sine subtraction formula, is:
sin(15) = (√6 – √2) / 4