Question 669251
what is cos(2theta) if cos=1/3 and csc<0
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csc<0 in quadrants III and IV
cos>0 in quadrants I and IV
so, we are working with reference angle in quadrant IV
cosx=1/3=adjacent side/hypotenuse in quadrant IV where cos>0
opposite side=&#8730;(3^2-1^2)=&#8730;(9-1)=&#8730;8
sinx=opposite side/hypotenuse=-&#8730;8/3(in quadrant IV where sin<0
cos2x=cos^2x-sin^2x=1/9-8/9=-7/9
Check:
cos^-1=1/3
x&#8776;70.529º
2x&#8776;141.05756º
cos2x&#8776;-0.7777..
-7/9&#8776;-0.7777..