Given that tanx=4/11 and secx<0,find cos(2x)

Answer 1

tanx = 4/11 tanx>0 means x is in quadrant 1 or 3

secx<0 means x is in quadrant 2 or 3

so x is in quadrant 3

tanx = 4/11 = opposite/adjacent with hypotenuse = sqr(16+121) =sqr137

cosx = adjacent/hypotenuse = 11/-sqr137, sinx = opposite/hypotenuse = 4/-sqr137

cos2x = cos^2(x)-sin^2(x)= 121/137 - 16/137= (121-16)/137 = 105/137

cos2x =105/137= about 0.7664234

x is in quadrant 3,

2x is in quadrant 1

check the answer using a calculator with inverse trig functions

x= arctan(4/11)= about 19.983 degrees or 180+19.983 = 199.983

2x=39.966 or 399.966 degrees

cos2x=cos39.966 = about .766426 or cos399.966 =cos(399.966-360) = cos39.966 = .766426

close enough

actually the secant sign is irrelevant to finding cos2x,but not to finding x or 2x

Answer 2