Question 526717: if cos theta = -2/3 and 450 degrees < theta < 540degrees, find
a. the exact value of cos (1/2) theta
b. the exact value of tan 2theta
Answer by lwsshak3(11628)   (Show Source):
You can put this solution on YOUR website! if cos theta = -2/3 and 450 degrees < theta < 540degrees, find
a. the exact value of cos (1/2) theta
b. the exact value of tan 2theta
**
using x in place of theta
450º < x < 540º places the angle in quadrant II where cos<0
cos x=-2/3 means you are working with a right triangle whose adjacent side is= -2, and the hypotenuse=3. By the Pythagorean Theorem, the opposite side=√5.
Angle x=arccos(-2/3)+360º=131.81º+360=491.81º
Reference angle=540-491.81=48.19º
..
a. cos x/2=±√[(1+cosx)/2]=-√[(1-2/3)/2]=-√(1/6)=-1/√6=-√6/6=-0.408..
check: cos(x/2)=cos(491.81/2)=cos(245.91º)=-0.408..
..
b. tan2x=(2 tanx)/(1-tan^2x)
tan x=√5/-2 (opposite side/adjacent side)
tan^2x=5/4
tan2x==2*(√5/-2)/(1-5/4)=-√5/(-1/4)=4√5=8.94..
tan 2x=tan(2*491.81)=tan(983.62º)=8.94..
..
Ans:
a. exact value of cos (1/2) theta=-√6/6
b. exact value of tan 2theta =4√5
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