Instead of typing out the half angle formulas, here is some artwork that spells them out nicely. Tangent is sine divided by cosine, so you would expect the square root sign to be involved, but a little algebraic manipulation makes it go away, as we will see in class Wednesday.
It all depends on whether we multiply the fraction under the square root sign by (1+ cosa)/(1 + cosa) or (1 - cosa)/(1 - cosa).
Here are some practice problems. Assume the angles are in the first quadrant, which means that half the angle must also be in the first quadrant, so all the major trig functions are positive.
a) cosa = 1/3
b) cosb = 1/4
c) cosc = 1/5
Answers in the comments.
a) cosa = 1/3
ReplyDeletecos(a/2) = sqrt((4/3)/2) = sqrt(4/6) = sqrt(2/3)= sqrt(6)/3
sin(a/2) = sqrt((2/3)/2) = sqrt(2/6) = sqrt(1/3)= sqrt(3)/3
tan(a/2) = (2sqrt(2)/3)/(4/3) =
sqrt(2)/2
b) cosb = 1/4
cos(b/2) = sqrt((5/4)/2) = sqrt(5/8) = sqrt(10/16)=
sqrt(10)/4
sin(b/2) = sqrt((3/4)/2) = sqrt(3/8) = sqrt(6/16)=
sqrt(6)/4
tan(b/2) = (sqrt(15)/4)/(5/4) =
sqrt(15)/5
c) cosc = 1/5
cos(c/2) = sqrt((6/5)/2) = sqrt(6/10) = sqrt(3/5) =
sqrt(15)/5
sin(c/2) = sqrt((4/5)/2) = sqrt(2/5) =
sqrt(10)/5
tan(c/2) = (sqrt(10)/5)/(sqrt(15)/5) = sqrt(2/3)
sqrt(6)/3