Question

Find all solutions of the equation in the interval [0, 2π) algebraically.

sin^2 x + cos x + 1 = 0

Answer 1

\(\bf \textit{recall that }\color{blue}{sin^2(\theta)+cos^2(\theta)=1\implies sin^2(\theta)=1-cos^2(\theta)}\qquad thus\\ \quad \\

sin^2(x)+cos(x)+1=0\implies [1-cos^2(x)]+cos(x)+1=0\\ \quad \\
-cos^2(x)+cos(x)+2=0\)

solve like you'd any quadratic

Answer 2

and how would i do that?

Answer 3

well... use the quadratic formula at most

other than that, just factor it, notice that it's just \(\bf -cos^2(x)+cos(x)+2=0\qquad cos(x) = a\\ \quad \\
-a^2+a+2=0\)

Answer 4

huh?