How do you find the exact value of sin 5pi/8?
cos2θ = 1 - 2sin^2 θ
since 2θ = 5pi/4, cos 2θ = -1/√2
-1/√2 = 1 - 2sin^2 5π/8
2sin^2 5π/8 = 1 + 1/√2
sin 5π/8 = √(1 + 1/√2)/2
= √(2+√2) / 2
You are talking about 112.5 degrees
that is 90 + 22.5
22.5 is half of 45
so draw this in the third quadrant
I know functions of 45 degrees
sin 45 = +1/sqrt 2
cos 45 = +1/sqrt 2
so what are the functions of 22.5 degrees?
sin(45/2) = +/- sqrt[(1-cos 45)/2]
cos (45/2) = +/-sqrt[(1+cos 45)/2]
in this case we want the negative cos for the sin (look at sketch) because in quadrant 3
so sin 112.5 = -sqrt [ (1+1/sqrt 2)/2]
= -sqrt [ (1 + sqrt 2)/2 sqrt 2
=- sqrt [ (2+sqrt2)/4 ]
= - .923
check sin 112.5 on calculator = -.923
sin is positive in QII
whoops, sorry
To find the exact value of sin(5π/8), we can use the half-angle formula for sine.
The half-angle formula for sine states that sin(x/2) = ±√[(1 - cos(x))/2].
In this case, x = 5π/4, so we want to find sin(5π/8).
1. First, we need to find cos(5π/4).
To find cos(5π/4), we use the formula cos(x) = ±√[(1 + cos(2x))/2].
Let's find cos(10π/8) since it's double the angle we need here.
cos(10π/8) = ±√[(1 + cos(2 * (5π/8)))/2] = ±√[(1 + cos(5π/4))/2]
Notice that cos(5π/4) is the value we are looking for in the first place.
So, cos(10π/8) = ±√[(1 + cos(5π/4))/2] = ±√[(1 + cos(5π/4))/2]
Rearrange the equation to isolate cos(5π/4):
cos(5π/4) = 2 * cos(10π/8) - 1
2. Now we can find sin(5π/8) using the half-angle formula for sine.
sin(5π/8) = ±√[(1 - cos(5π/4))/2]
Substitute the value of cos(5π/4) we found in step 1:
sin(5π/8) = ±√[(1 - (2 * cos(10π/8) - 1))/2]
Simplify the expression:
sin(5π/8) = ±√[(2 - 2 * cos(10π/8))/2]
sin(5π/8) = ±√[(2 - 2 * cos(5π/4))/2]
sin(5π/8) = ±√[(1 - cos(5π/4))]
sin(5π/8) = ±√[2(1 - cos(5π/4))/2]
sin(5π/8) = ±√[1 - cos(5π/4)]
Finally, we can substitute the value of cos(5π/4) we found earlier:
sin(5π/8) = ±√[1 - (2 * cos(10π/8) - 1)]
sin(5π/8) = ±√[1 - (2 * cos(5π/4) - 1)]
sin(5π/8) = ±√[2 * cos(5π/4)]
sin(5π/8) = ±√[2 * (-√2/2)]
sin(5π/8) = ±√[-√2]
sin(5π/8) = ±(-√2) or ±√2
Therefore, the exact value of sin(5π/8) is ±√2.