How do I find the exact value of sin (pi/24)?
I am not under standing could you show me ? to tell me ?
I applied the half angle formula for sin and I got the square root of 2 minus the square root of 3 divided by 2 . and for cos I got thesqurof 2 plus thesqr of3divided by 2 .
I really need help . I find this hard Steve .
Do I subtract ?
sin π/6 = 1/2
cos π/6 = √3/2
so,
cos π/12 = √(1+√3/2) / 2
sin π/24 = √(1-cos π/12) / 2
= 1/2 √(1 - (√(1+√3/2) / 2))
= 1/2√2 √(2 - √(1+√3/2))
= 1/4 √(2√2 - √(1+√3))
...
To find the exact value of sin(pi/24), we can use the sum-to-product identities and the half-angle identity. Here's how you can do it step-by-step:
Step 1: Start with the angle pi/12, which is the half of pi/6.
sin(pi/12) = sqrt(2 - sqrt(3)) / 2
Step 2: Use the double-angle identity for sine to find sin(pi/6). Recall that the double-angle identity for sine is sin(2x) = 2sin(x)cos(x).
sin(pi/6) = 2 * sin(pi/12) * cos(pi/12)
Step 3: Now we need to find cos(pi/12) using the half-angle formula for cosine. The formula is cos(x/2) = sqrt((1 + cos(x))/2).
cos(pi/12) = sqrt((1 + cos(pi/6)) / 2)
Step 4: Substitute the values from step 1 and 3 into step 2:
sin(pi/6) = 2 * (sqrt(2 - sqrt(3)) / 2) * sqrt((1 + sqrt(3)/2) / 2)
Step 5: Simplify and evaluate:
sin(pi/6) = sqrt(2 - sqrt(3)) * sqrt(1 + sqrt(3)/2)
Step 6: Now we need to find sin(pi/3) to use the sum-to-product identities. Recall that sin(pi/3) = sqrt(3)/2.
Step 7: Use the sum-to-product identities for sine to find sin(pi/8). The sum-to-product identity is sin(A) + sin(B) = 2*cos((A - B)/2)*sin((A + B)/2).
sin(pi/8) = sin(pi/12 + pi/6) = 2 * cos((pi/6 - pi/12)/2) * sin((pi/6 + pi/12)/2)
= 2 * cos(pi/4) * sin(pi/4)
= 2 * (sqrt(2)/2) * (sqrt(2)/2)
= 1
Step 8: Use the sum-to-product identity again to find sin(pi/24). The identity is sin(A) = sin(A + B) - sin(B).
sin(pi/24) = sin(pi/12 + pi/8) - sin(pi/8)
= sin(pi/12 + pi/8) - 1
Step 9: Substitute the values from step 1 and 7 into step 8:
sin(pi/24) = sqrt(2 - sqrt(3)) * sqrt(1 + sqrt(3)/2) - 1
Finally, you can simplify and evaluate the expression to find the exact value of sin(pi/24).
sin π/6 = 1/2
cos π/6 = √3/2
sin(x/2) = √(1-cosx) / 2
cos(x/2) = √(1+cosx) / 2
apply the half-angle formula twice to get
sin π/24 = 1/2 √(2-√(2+√3))