Find every angle theta with 0 <= theta <= 2pi radians, and 2 sin^2(theta)+cos(theta)=2

Find every angle theta with 0 <= theta <= 2pi radians, and 2 sin^2(theta)+cos(theta)=2

Question
Find every angle theta with \(\displaystyle{0}\le\theta\le{2}\pi{r}{a}{d}{i}{a}{n}{s},{\quad\text{and}\quad}{2}{{\sin}^{{2}}{\left(\theta\right)}}+{\cos{{\left(\theta\right)}}}={2}\)

Answers (1)

2021-02-26
\(\displaystyle{{\sin}^{{2}}=}{1}-{{\cos}^{{2}},}\)
so using x as \(\displaystyle\theta,{2}-{2}{{\cos}^{{2}}{x}}+{\cos{{x}}}={2},\)
\(\displaystyle{2}{{\cos}^{{2}}{x}}-{\cos{{x}}}={0},{\cos{{x}}}{\left({2}{\cos{{x}}}-{1}\right)}={0},\)
\(\displaystyle{\cos{{x}}}={0}{\quad\text{or}\quad}{0.5}\)
\(\displaystyle{x}=\frac{{\pi}}{{3}}{\quad\text{or}\quad}{60}°,\frac{{\pi}}{{2}}{\quad\text{or}\quad}{90}°,{3}\frac{{\pi}}{{2}}{\quad\text{or}\quad}{270}°,{5}\frac{{\pi}}{{3}}{\quad\text{or}\quad}{300}°.\)
0

Relevant Questions

asked 2020-11-12
If \(\displaystyle{\cot{{\left(\theta\right)}}}={7}\), what is \(\displaystyle{\sin{{\left(\theta\right)}}},{\cos{{\left(\theta\right)}}},{\sec{{\left(\theta\right)}}}\) between 0 and \(\displaystyle{2}\pi\)?
asked 2021-02-25
Solve the equation \(\frac{\sin^{2}\theta/}{cos \theta}= \sec \theta-\cos \theta\)
asked 2021-01-19
Interval \([0,2pi)\)
\(2 cos^2 x - cos x = 0\)?
asked 2021-01-19
Find an equation of the tangent line to the curve at the given point.
\(\displaystyle{y}={\sin{{\left({\sin{{x}}}\right)}}},{\left({2}\pi,{0}\right)}\)
asked 2021-01-31
If \(\cot \theta= -\sqrt{3} \ \text{and} \ sec \theta<0\)
Find \(\sin \theta\)
asked 2021-01-07
Prove that \(\displaystyle{\sec{{\left(\theta\right)}}}+{\csc{{\left(\theta\right)}}}={\left({\sin{{\left(\theta\right)}}}+{\cos{{\left(\theta\right)}}}\right)}{\left({\tan{{\left(\theta\right)}}}+{\cot{{\left(\theta\right)}}}\right)}\)
asked 2021-02-01
Solve the equation \(\sin \theta \cot \theta = \cos \theta\)
asked 2020-11-23
Solve the equation \(\frac{1}{\csc \theta-\cos \theta}=\frac{1+\cos \theta}{\sin \theta}\)
asked 2020-11-12
Solve the equation \(\frac{(2\sin \theta \sin 2\theta)}{\cos \theta+\cos 3\theta}=\tan (\theta) \tan(2\theta)\)
asked 2020-12-15
Solve the equation on the interval [0,2pi] \(\sin{{\left({x}+\frac{\pi}{{4}}\right)}}+ \sin{{\left({x}-\frac{\pi}{{4}}\right)}}={1}\)
...