Ask question

# What is the solution of cos2x - cos x = 0 text{in the interval} [0, 2pi ) ? # What is the solution of cos2x - cos x = 0 text{in the interval} [0, 2pi ) ?

Question
Trigonometric equation and identitie asked 2021-02-13
What is the solution of $$\cos2x - \cos x = 0\ \text{in the interval}\ [0, 2\pi )$$ ?

## Answers (1) 2021-02-14
Let use the trigoniometric identity $$\cos2x = 2\cos^{2}x - 1:$$
$$2\cos^{2}x - 1 - \cos x = 0$$
Factorize: $$(\cos x - 1) (2\cos x + 1) = 0$$
Use the zero product property: $$\cos x - 1 = 0\ or\ 2\cos x + 1 = 0$$
Solve both equations to consine: $$\cos x = 1\ or\ \cos x = -\frac{1}{2}$$
$$\cos x = 1$$ when $$x = 0$$
$$x = -\frac{1}{2}$$ when $$x = \pi - \pi/3 = (2\pi)/3\ or\ x = \pi + \pi/3 = (4\pi)/3$$
Aswer: $$x = 0, (\frac{2\pi}{3}), (4\pi)/3$$

### Relevant Questions asked 2021-03-09

Show by substitution that $$u(x,t)=\cos(απx)e^−α^2π^2t$$ is a solution of the heat equation $$ut=uxx$$ on any interval [0, L]. asked 2021-02-27
Solve the equations and inequalities. Write the solution sets to the inequalities in interval notation. $$\displaystyle{\left({x}^{{2}}-{9}\right)}^{{2}}-{2}{\left({x}^{{2}}-{9}\right)}-{35}={0}$$ asked 2021-04-15
A car initially traveling eastward turns north by traveling in a circular path at uniform speed as in the figure below. The length of the arc ABC is 235 m, and the car completes the turn in 33.0 s. (Enter only the answers in the input boxes separately given.)
(a) What is the acceleration when the car is at B located at an angle of 35.0°? Express your answer in terms of the unit vectors $$\displaystyle\hat{{{i}}}$$ and $$\displaystyle\hat{{{j}}}$$.
1. (Enter in box 1) $$\displaystyle\frac{{m}}{{s}^{{2}}}\hat{{{i}}}+{\left({E}{n}{t}{e}{r}\in{b}\otimes{2}\right)}{P}{S}{K}\frac{{m}}{{s}^{{2}}}\hat{{{j}}}$$
(b) Determine the car's average speed.
3. ( Enter in box 3) m/s
(c) Determine its average acceleration during the 33.0-s interval.
4. ( Enter in box 4) $$\displaystyle\frac{{m}}{{s}^{{2}}}\hat{{{i}}}+$$
5. ( Enter in box 5) $$\displaystyle\frac{{m}}{{s}^{{2}}}\hat{{{j}}}$$ asked 2021-01-19
Interval $$[0,2pi)$$
$$2 cos^2 x - cos x = 0$$? asked 2020-11-10
Solve $$\frac{\csc x(sin^{2}x+cos^{2}x\tan x)}{\sin x+\cos x}$$ asked 2021-03-08
Solve $$(\sin x + \cos x)(\sin x + \cos x)$$ asked 2021-02-08
Explain how we can evaluate the expression $$\displaystyle{\cos{{\left({\arctan{{\left({\frac{{{v}}}{{{a}}}}\right)}}}\right)}}}$$ and what the evaluation is in terms of v and a. asked 2021-02-23
Solid NaBr is slowly added to a solution that is 0.010 M inCu+ and 0.010 M in Ag+. (a) Which compoundwill begin to precipitate first? (b) Calculate [Ag+] when CuBr justbegins to precipitate. (c) What percent of Ag+ remains in solutionat this point?
a) AgBr: $$\displaystyle{\left({0.010}+{s}\right)}{s}={4.2}\cdot{10}^{{-{8}}}$$ $$\displaystyle{s}={4.2}\cdot{10}^{{-{9}}}{M}{B}{r}$$ needed form PPT
CuBr: $$\displaystyle{\left({0.010}+{s}\right)}{s}={7.7}\cdot{\left({0.010}+{s}\right)}{s}={7.7}\cdot{10}^{{-{13}}}$$ Ag+=$$\displaystyle{1.8}\cdot{10}^{{-{7}}}$$
b) $$\displaystyle{4.2}\cdot{10}^{{-{6}}}{\left[{A}{g}+\right]}={7.7}\cdot{10}^{{-{13}}}$$ [Ag+]$$\displaystyle={1.8}\cdot{10}^{{-{7}}}$$
c) $$\displaystyle{\frac{{{1.8}\cdot{10}^{{-{7}}}}}{{{0.010}{M}}}}\cdot{100}\%={0.18}\%$$ asked 2021-03-26
Solve the equations and inequalities. Find the solution sets to the inequalities in interval notation. $$\displaystyle{3}{x}{\left({x}-{1}\right)}={x}+{6}$$ asked 2021-03-04
Solve $$\frac{(\sin \theta+\cos \theta)}{\cos \theta}+\frac{(\sin \theta-\cos \theta)}{\cos \theta}$$
...