Question

Find the sum of all possible values of θ (in degrees) restricted to the domain 0≤θ≤360∘, satisfying

sin4θ+sin2θcos2θ+cos4θ=3/4

Answer 1

help

Answer 2

Observe the following:\[\bf \sin(2\theta)=2\sin(\theta)\cos(\theta) \implies \sin(4\theta)=2\sin(2\theta)\cos(2\theta)\]Must this substitution we get:\[\bf \sin(4\theta)+\sin(2\theta)\cos(\theta)+\cos(4\theta)=3/4 \]\[\bf \implies 2\sin(2\theta)\cos(2\theta)+\sin(2\theta)\cos(2\theta)+\cos(4\theta)=3/4 \]Simplifying yields:\[\bf 3\sin(2\theta)\cos(2\theta)+\cos(4\theta)=3/4\]Now we can change cos(4x) through the following identitiy:\[\bf \cos(2\theta)=1-2\sin^2(\theta) \implies \cos(4\theta)=1-2\sin^2(2\theta)\]Making this substituion we now get:\[\bf 3\sin(2\theta)\cos(2\theta)+(1-2\sin^2(2\theta))=3/4 \]\[\bf \implies 3\sin(2\theta)\cos(2\theta)-2\sin^2(2\theta)+1/4 =0\]I have to leave but use wolframalpha.com if you need to confirm your solution. @fabsam14

Answer 3

omg this is amazing are u some kind of wizard c: @genius12

Answer 4

@genius12