Give exact answers
Sin(11pi/8)
so this is sin(pi/4+pi/3)
sin(pi/4)*cos(pi/3)+cos(pi/3)*sin(pi/4)
rad2/2*1/2+1/2*rad2/2
which gives me rad2+rad2 all over 4 ...i don't know y im getting it wrong
pi/4 + pi/3 does NOT equal 11 pi/8
It equals 7 pi/12
sin (11pi/8) = -sin3pi/8
= -sin[(1/2)3pi/4]
Now use the "sine half angle" formula
= -sqrt[(1/2)(1-cos3pi/4)]
= -sqrt[(1/2)(1+(sqrt2)/2)]
To find the value of sin(11π/8), we can break it down as sin(π/4 + π/3) and then use trigonometric identities.
Starting with sin(π/4 + π/3):
sin(π/4 + π/3) = sin(π/4) * cos(π/3) + cos(π/3) * sin(π/4)
Now, let's evaluate sin(π/4):
sin(π/4) = √2/2
Next, let's evaluate cos(π/3):
cos(π/3) = 1/2
Substituting the values back into the equation:
sin(π/4 + π/3) = (√2/2) * (1/2) + (1/2) * (√2/2)
Now, let's simplify:
(√2/2) * (1/2) + (1/2) * (√2/2) = √2/4 + √2/4
Combining like terms:
√2/4 + √2/4 = 2√2/4
Simplifying further:
2√2/4 = √2/2
Therefore, sin(11π/8) = √2/2.
Let's break down the steps to find the exact value of sin(11π/8):
1. Start with sin(11π/8).
2. Rewrite the angle using the sum-of-angle identity: 11π/8 = π/4 + π/3.
3. Apply the sum-of-angle formula for sine:
sin(π/4 + π/3) = sin(π/4)cos(π/3) + cos(π/4)sin(π/3).
4. Substitute the values of sin(π/4) and cos(π/3):
(sqrt(2)/2)(1/2) + (1/2)(sqrt(3)/2).
5. Simplify the expression inside each parentheses:
sqrt(2)/4 + sqrt(3)/4.
6. Combine the two terms:
(sqrt(2) + sqrt(3))/4.
So, the exact value of sin(11π/8) is (sqrt(2) + sqrt(3))/4.