Search 75,745 tutors
FIND TUTORS
Ask a question
0 0

Use the half angle formulas to determine.

Tutors, please sign in to answer this question.

2 Answers

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
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