Use the half angle formulas to determine.

Question

A. sin22.5 degrees

and 

B. tanpi/12 degrees

Answer 1

A) sin(22.5)degrees=sin(45/2)degrees

sin(45/2)=+sqrt[(1-cos(45))/2]

             

             =+sqrt[(1-1/sqrt(2))/2]

         

             =+sqrt[{(sqrt(2)-1)/sqrt(2)}/2]

             =+sqrt[{2-sqrt(2)/2}/2]

             =+sqrt[{2-sqrt(2)}/4]

             =+sqrt[{2-1.414213}/2

             =+sqrt{0.585786}/2

             =+0.765366/2

             =0.382683

 

B) tan pi/12 degrees

    1 degree=pi/180

    15pi/(15)(12)=15*(pi/180)=15 degrees

    tan15 degrees=0.26794919

 getting the answer using half-angle formulas gives us the following:

tan(30/2)=sqrt[(1-cos30)/(1+cos30)]

              =sqrt[(1-sqrt(3)/2)/(1+sqrt(3)/2)]

simplify 1-sqrt(3)/2 and 1+sqrt(3)/2

1-sqrt(3)/2=[2-sqrt(3)]/2

1+sqrt(3)/2=[2+sqrt(3)]/2

divide[2-sqrt(3)]/2 by [2+sqrt(3)]/2

we get [2-sqrt(3)]/[2+sqrt(3)]

rationalize the denominator

we get [2-sqrt(3)][2-sqrt(3)]/[2-sqrt(3)][2+sqrt(3)]

this gives us [4-4sqrt(3)+3]/[4-3] which simplifies to

[7-4sqrt(3)]

[7-4sqrt(3)]=7-4(1.7320508)=7-6.9282032=0.0717968

now we take the square root of 0.0717968

sqrt(0.0717968)=0.26794919 which we got before

tan(pi/12)=0.26794919

Answer 2

cos45=1/√2

1-2sin²22.5=cos45=1/√2

 

sin²22.5=(2-√2)/2

 

sin22.5=√(2-√2)/√2

 

2tan(pi/12)/(1-tan²(pi/12))=tan(pi/6)=1/√3

 

1-tan²(pi/12)=2√3tan(pi/12)

 

tan²(pi/12)+2√3tan(pi/12)-1=0

 

tan(pi/12)=2-√3