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Question

determine the exact value of sin (5π/8)

 

 

Answer 1

sin(5 pi/8) = sin( (5 pi/4)/2 )

 

sin( θ/2 ) = ± √( (1 – cos(θ) )/2 )

 

5 pi/8 lies in the 2nd Quadrant where the sine is positive.

 

So sin( (5 pi/4)/2 ) = + √( (1 – cos(5 pi/4) )/2 )

 

5 pi/4 lies in the 3rd Quadrant where cosine is negative. Its reference angle is pi/4 = 45°. From the Special Triangle 45°-45°-90° we know that cos(45°) = √(2)/2.

 

So sin(5 pi/8) = + √( (1 – (–√(2)/2) )/2 ),

 

sin(5 pi/8) = √( (1 + √(2)/2 )/2 ),

 

sin(5 pi/8) = 1/2 √( 2 + √(2) )

Answer 2

 

    5∏/ 8 = 4∏/ 8 + ∏/8

 

   Sin ∏/4 = 2 Sin ∏/8 . Cos ∏/8

 

  1/2  = 4  sin² ∏/8 ( 1 - sin² ∏ /8)

 

      8 Sin² ∏/8 - 8 Sin⁴ ∏/8 - 1 =0

 

 

      Sin² ∏/8 = 8/16 ± √(64 +32)/-16 

 

        Sin ² ∏/8 = 1/2 - √2 /2 / acceptable answer

 

        

        Cos² ∏/ 8 =   1 - Sin² ∏/8 = 1 - ((1/2 -√2/2) )^2= 

 

Sin 5∏/8 = Sin ∏/2 Cos∏/8 + Cos∏/2 Sin∏/8

             

              = 1 . Cos∏/8 +  0 .Sin ∏/8= Cos ∏ /8

       

Cos ∏/8 = sin 5∏/8 = √( 1/2 + √2)/2

Answer 3

let's start by the following formula:

 

sin(Π/2+θ) =cos(θ)

 

now we'll simplify the original equation into the above format:

 

sin(5Π/8) = sin(4Π/8 + Π/8) = sin (Π/2 + Π/8) = cos(Π/8)

 

another formula is:

 

cos(θ) = √(1+cos(2θ))/2

 

therefore:

 

cos(Π/8) =  √((1+cos(Π/4))/2) = √((1+√2/2)/2) = √((2+√2)/4) = √(2+√2)/2

 

so,

 

sin(5Π/8) = √(2+√2)/2