Find all values of x

Question

1.) Find all values of X that satisfy the equation sin^2(x)=2cos(x)+2 in the interval 0≤x<2pi

 

 

 

 

 

2.) Find all values of x in the interval 0≤x<2pi that satisfy the equation tan(x)= - sqrt(3)/(3)

 

Answer 1

sin^2x=2cosx+2

sin^2x+cos^2x=1

sin^2x=1-cos^2x

substitute

1-cos^2x=2cosx+2

cos^2x+2cosx+1=0

factor

(cosx+1)(cosx+1)=0

cosx+1=0

cosx=-1

look on the unit circle

x=180º

 

tanx=-√3/3

tan=sin/cos

look on the unit circle at 5∏/6=150º and at 11∏/6=330º

in each case you have (1/2)/(-√3/2) and (-1/2)/(√3/2)

they both equal (1/2)(-2/√3)=(-1/√3)=-√3/3

x=150º and x=330º