Question 672645: Use the given information to find cos(x/2), sin(x/2), and tan(x/2):
tan x = 2, pi < x < 3pi/2
Here is my solution, but I don't know how to get sin(x/2) and tan(x/2):
1 + (2)^2 = sec^2x
5 = sec^2x
sec x = -sqrt(5)
cos x = -1 / sqrt(5) = -sqrt(5)/5
cos^2(x/2) = 1/2(1+(-sqrt(5)/5))
= 5-(sqrt(5)/10)
cos(x/2) = -sqrt(5-sqrt(5)/10)
sin^2(x/2) = 1/2(1-(-sqrt(5)/5))
= 5+(sqrt(5)/10)
sin(x/2) = ?
tan(x/2) = ?
Answer by lwsshak3(11628)   (Show Source):
You can put this solution on YOUR website! Use the given information to find exact values of cos(x/2), sin(x/2), and tan(x/2):
tan x = 2, pi < x < 3pi/2
**
Given information shows that x is in quadrant III, where cos<0, sin<0, tan>0
tanx=2=opp side/adj side=2/1
hypotenuse=√(2^2+1^2)=√(4+1)=√5
sinx=opp side/hypotenuse=-2/√5
cosx=adj side/hypotenuse=-1/√5
..
use half-angle identities to solve
cos(x/2)=±[√(1+(cosx)/2)]
=-[√(1-(1/√5)/2)]
=-[√(√5-1)/2√5)]
..
sin(x/2)=±[√(1-(cosx)/2)]
=-[√(1+(1/√5)/2)]
=-[√(√5+1)/2√5)]
..
tan(x/2)=sinx/(1+cosx)
=-2√5/(1-(1/√5)
=-2√5/(√5-1)/√5
=-2/√5-1
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