Question

I just need help with the second part. Please help!!!

Answer 1

`\large\boxed{x\approx35^0\ \vee\ x\approx145^o\ \vee\ x\approx215^o\ \vee\ x\approx325^o}`

Explanation:

`\text{From (i) we have}\ \cos^4x=1-2\sin^2x+\sin^4x.\\\\\text{Substitute to the equation}\ 8\sin^4x+\cos^4x=2\cos^2x:\\\\8\sin^4x+1-2\sin^2x+\sin^4x=2\cos^2x\\\\\footnotesize{\text{Use}\ \sin^2x+\cos^2x=1\to\cos^2x=1-\sin^2x}\\\\9\sin^4x+1-2\sin^2x=2(1-\sin^2x)\\\\\footnotesize{\text{use the distributive property}\ a(b+c)=ab+ac}\\\\9\sin^4x+1-2\sin^2x=2-2\sin^2x\\\\_{\text{add}\ 2\sin^2x\ \text{from both sides}}\\\\9\sin^4x+1=2`

`\footnotesize{\text{subtract 1 from both sides}}\\\\9\sin^4x=1\\\\\footnotesize{\text{divide both sides by 9}}\\\\\sin^4x=(1)/(9)\to\sin x=\pm\sqrt[4]{(1)/(9)}\\\\\sin x=\pm(1)/(\sqrt3)\cdot(\sqrt3)/(\sqrt3)\\\\\sin x=\pm(\sqrt3)/(3)\\\\\text{look at the picture}`