find the exact value by using the appropriate half-angle identity:
sin(5pi/12)
Some students can "think" better in degrees than radians, so 5π/12 rd = 75°
sin75 = sin(30+45)
= sin30cos45 + cos30sin45
= (1/2)√2/2 + (√3/2)(√2/2)
= (√2 + √6)/4
To find the exact value of sin(5pi/12) using the appropriate half-angle identity, we can start by using the half-angle formula for sine:
sin(x/2) = ± √[(1 - cos(x))/2]
In this case, x is equal to 5pi/6, so let's substitute it into the formula:
sin((5pi/6)/2) = ± √[(1 - cos(5pi/6))/2]
Now, we need to find the value of cos(5pi/6). To do this, we can use the cos(x) = sin(π/2 - x) identity:
cos(x) = sin(π/2 - x)
Therefore, cos(5pi/6) is equal to sin(π/2 - 5pi/6):
cos(5pi/6) = sin(pi/2 - 5pi/6) = sin(pi/3) = √3/2
Now, let's substitute the value of cos(5pi/6) into the half-angle formula:
sin((5pi/6)/2) = ± √[(1 - √3/2)/2]
Simplifying further:
sin((5pi/6)/2) = ± √[(2 - √3)/4]
So, sin(5pi/12) is equal to ± √[(2 - √3)/4].