How do you solve using CAST: sin 2θ +cos θ = 0 between 0 and 2pi?

1 Answer
Jim G.
Apr 27, 2018

#theta=pi/2,(7pi)/6,(3pi)/2,(11pi)/6#

Explanation:

#"using the "color(blue)"trigonometric identity"#

#•color(white)(x)sin2theta=2sinthetacostheta#

#rArr2sinthetacostheta+costheta=0#

#"take out "color(blue)"common factor "costheta#

#rArrcostheta(2sintheta+1)=0#

#"equate each factor to zero and solve for "theta#

#costheta=0rArrtheta=pi/2,(3pi)/2#

#2sintheta+1=0rArrsintheta=-1/2#

#"Since "sintheta<0" then "theta" is in third or fourth"#
#"quadrant"#

#theta=sin^-1(1/2)rArrtheta=pi/6larrcolor(red)"related acute angle"#

#rArrrArrtheta=(pi+pi/6)=(7pi)/6larrcolor(red)"third quadrant"#

#rArrtheta=(2pi-pi/6)=(11pi)/6larrcolor(red)"fourth quadrant"#

#rArrtheta=pi/2,(7pi)/6,(3pi)/2,(11pi)/6to(0,2pi)#

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