To find the exact value of sin(5π/12), we can use the half-angle identity. The half-angle formula states:
sin(θ/2) = ±√((1 – cos(θ)) / 2)
First, let’s rewrite 5π/12 in a way that our half-angle formula can be applied. Notice that:
5π/12 = π/3 + π/4
Now, we can find sin(5π/12) as:
sin(5π/12) = sin(π/3 + π/4)
Using the sine addition formula, we have:
sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
Here, a = π/3 and b = π/4.
Calculating sin(π/3) and cos(π/3):
- sin(π/3) = √3/2
- cos(π/3) = 1/2
Now for sin(π/4) and cos(π/4):
- sin(π/4) = √2/2
- cos(π/4) = √2/2
Plugging these values back into the sine addition formula:
sin(5π/12) = sin(π/3)cos(π/4) + cos(π/3)sin(π/4)
Substituting the values gives:
sin(5π/12) = (√3/2)(√2/2) + (1/2)(√2/2)
This simplifies to:
sin(5π/12) = (√6/4) + (√2/4)
Thus, combining these terms results in:
sin(5π/12) = (√6 + √2) / 4
So the exact value of sin(5π/12) is (√6 + √2) / 4.