find an exact value for sin (3pi/8)
Remark :
_________________________________
pi / 8 = 22.5° = 22° 30'
3 pi / 8 = 4 pi / 8 - pi / 8 = pi / 2 - pi / 8
3 pi / 8 = 90° - 22° 30' = 67° 30'
pi / 4 = 45°
sin ( pi / 4 ) = sqroot ( 2 ) / 2
cos ( pi / 4 ) = sqroot ( 2 ) / 2
pi / 2 = 90°
sin ( pi / 2 ) = 1
cos ( pi / 2 )= 0
__________________________________
3 pi / 8 = pi / 2 - pi / 8
sin ( A - B ) = sin ( A ) cos ( B ) - cos ( A ) sin ( B )
sin ( 3 pi / 8 ) =
sin ( pi / 2 - pi / 8 ) =
sin ( pi / 2 ) cos ( pi / 8 ) - cos ( pi / 2 ) sin ( pi / 8 ) =
1 * cos ( pi / 8 ) - 0 * sin ( pi / 8 ) =
cos ( pi / 8 )
So :
sin ( 3 pi / 8 ) = cos ( pi / 8 )
pi / 8 = ( pi / 4 ) / 2
cos ( A / 2 ) = + OR - sqroot [ ( 1 + cos A ) / 2 ]
cos ( pi / 8 ) = cos ( pi / 4 ) / 2 = + OR - sqroot [ ( 1 + cos ( pi / 4 ) ) / 2 ] =
+ OR - sqroot [ ( 1 + sqroot ( 2 ) / 2 ) / 2 ] =
+ OR - sqroot [ ( 2 / 2 + sqroot ( 2 ) / 2 ) / 2 ] =
+ OR - sqroot [ ( ( 2 + sqroot ( 2 ) ) / 2 ) / 2 ] =
+ OR - sqroot [ ( 2 + sqroot ( 2 ) ) / ( 2 * 2 ) ] =
+ OR - sqroot [ ( 2 + sqroot ( 2 ) ) / 4 ] =
+ OR - sqroot [ 2 + sqroot ( 2 ) ] / sqroot ( 4 ) =
+ OR - sqroot [ 2 + sqroot ( 2 ) ] / 2
3pi / 8 is in quadrant I
In quadrant I cosine in positive so :
cos ( pi / 8 ) = + sqroot [ 2 + sqroot ( 2 ) ] / 2 = sqroot [ 2 + sqroot ( 2 ) ] / 2
We alredy know :
sin ( 3 pi / 8 ) = cos ( pi / 8 ) so :
sin ( 3 pi / 8 ) = sqroot [ 2 + sqroot ( 2 ) ] / 2
P.S.
If you do not know how to write this value go to:
w o l f r a m a l p h a . c o m
When page be open in rectangle type :
sin ( 3 pi / 8 )
then in rectangle click option =
To find the exact value of sin(3π/8), you can make use of the half-angle identity for sine.
The half-angle identity for sine states that sin(x/2) = ±√((1 - cos(x)) / 2).
In this case, x = 3π/4. So, sin(3π/8) = sin((3π/4)/2).
Let's apply the half-angle identity:
sin((3π/4)/2) = ±√((1 - cos(3π/4)) / 2)
We know that cos(3π/4) = -√2/2.
Therefore:
sin((3π/4)/2) = ±√((1 - (-√2/2)) / 2)
= ±√((1 + √2/2) / 2)
= ±√((2 + √2) / 4)
= ±√(2 + √2) / 2
So, sin(3π/8) = ±√(2 + √2) / 2, where the ± symbol indicates that there are two possible solutions for the positive and negative values of sine.
To find the exact value of sin(3π/8), we can use the half-angle identity for sine. The half-angle identity states that sin(θ/2) = ±√((1 - cos(θ))/2), where θ is the angle in radians.
In this case, we have θ = 3π/4, so we want to find sin(3π/8).
First, let's calculate the cosine of the angle by using the unit circle or a calculator:
cos(3π/4) = -√2/2
Now, we can substitute this value into the half-angle identity:
sin(π/4) = ±√((1 - cos(3π/4))/2)
sin(π/4) = ±√((1 + (√2/2))/2)
sin(π/4) = ±√((2 + √2)/4)
sin(π/4) = ±√(2 + √2)/2
Since sin(π/4) is positive in the first quadrant, we can take the positive value:
sin(3π/8) = √(2 + √2)/2
Therefore, the exact value of sin(3π/8) is √(2 + √2)/2.