help solve 2sin2x+sin(2x)+cos(2x)=0 exactly

Question

2sin2x+sin(2x)+cos(2x)=0

solve trigonometry equation exactly; 0°≤x<360°. 

Please and thank you!

Answer 1

I just did this in your first post. If you can't find it...

2sin^2x+sin2x+cos2x=0

sin2x=2sinxcosx is a trig identity

cos2x=cos^2x-sin^2x is a trig identity

substitute

2sin^2x+2sinxcosx+cos^2x-sin^2x=0

combine similar terms

sin^2x+2sinxcosx+cos^2x=0

factor this perfect square

(sinx+cosx)(sinx+cosx)=0

sinx+cosx=0

cosx=-sinx

divide both sides by cosx

1=-sinx/cosx

-1=sinx/cosx

sinx/cosx=tanx

substitute

-1=tanx

tanx=-1

tangent is negative in 2nd and 4th quadrants

tan45º=1 but we want -1

45º+90º=135º

tan135º=-1 (one answer)

45º+270º=315º (not 45+180 because we need to get to the 4th quadrant)

tan315º=-1 (second answer)

you can also look on the unit circle and find where sin and cos are opposites of each other because any number divided by its opposite is -1

this occurs at 3∏/4 (135º) and 7∏/4 (315º)