Question 691480: Find sin x/2, cos x/2, and tan x/2
from the given information.
sec x = 10/9 ,270° < x < 360°
sin x/2=
cos x/2=
tan x/2 =
Answer by lwsshak3(11628)   (Show Source):
You can put this solution on YOUR website! Find sin x/2, cos x/2, and tan x/2
from the given information.
sec x = 10/9 ,270° < x < 360°
sin x/2=
cos x/2=
tan x/2=
**
sec x=10/9=hypotenuse/adjacent side
hypotenuse=10
adjacent side=9
opposite side=√(10^2-9^2)=√(100-81)=√19
sin x=-√19/10 (in quadrant IV where sin<0)
cos x=9/10
..
Identities:
sin x/2=±√[(1-cos x)/2]
choose positive root because x/2 is in quadrant II where sin>0
sin x/2=√[(1-cos x)/2]=√[(1-9/10/2]=√.1/2=√.05
..
cos x/2=±√[(1+cos x)/2]
choose negative root because x/2 is in quadrant II where cos<0
cos x/2=-√[(1+cos x)/2]=-√[(1+9/10/2]=-√1.9/2=-√.95
..
tan x/2=sin x/(1+cos x)=-(√19/10)/(1+9/10)=-(√19/10)/1.9
..
How to check answers with calculator:
sec x=10/9
cos x=9/10
cos^-1=(9/10)≈25.84º (reference angle in specified quadrant IV)
standard position of angle=360-25.84=334.16
x/2=334.16/2=167.08
reference angle=180-167.08=12.92º
..
sin x/2=sin 12.92≈0.2236..(in quadrant II where sin>0)
√.05=0.2236..
..
you can check cos x/2 and tan x/2 in the same way
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