jelentetvq

2022-01-30

Solve the equation $\frac{\sqrt{3}}{2}\mathrm{sin}\left(x\right)-\mathrm{cos}x={\mathrm{cos}}^{2}x$

My approach${\mathrm{cos}}^{2}x=1-{\mathrm{sin}}^{2}x$

$\frac{\sqrt{3}}{2}\mathrm{sin}x-{\mathrm{cos}}^{2}x=\mathrm{cos}x$

$\frac{\sqrt{3}}{2}\mathrm{sin}x+{\mathrm{sin}}^{2}x-1=\mathrm{cos}x$

$\sqrt{3}\mathrm{sin}x+2{\mathrm{sin}}^{2}x-2=2\mathrm{cos}x$

Though the equation comes in form of$\mathrm{sin}x$ from here onward after squaring still not getting the answer.

My approach

Though the equation comes in form of

trnovitom06

Beginner2022-01-31Added 12 answers

A straightforward approach is indeed to square the equation:

$\frac{\sqrt{3}}{2}\mathrm{sin}x={\mathrm{cos}}^{2}x+\mathrm{cos}x$

and replace${\mathrm{sin}}^{2}x\text{}\text{by}\text{}1-{\mathrm{cos}}^{2}x$ . You obtain a quartic equation in cosx. Maybe not the most elegant solution, but you cannot go wrong with it. Also, one root is clear without calculation: $\mathrm{cos}x=\frac{1}{2}$ is fine. So you can surely reduce to a cubic equation. (And possibly even further to a quadratic one.)

and replace

vasselefa

Beginner2022-02-01Added 9 answers

Just to flesh out first answer, let $c=\mathrm{cos}x,\text{}s=\mathrm{sin}x$ so $\frac{s\sqrt{3}}{2}=c(1+c)$ and $3(1-{c}^{2})=4{c}^{2}{(1+c)}^{2}$ . After some rearrangment, $(c+1)(c-\frac{1}{2})(2{c}^{2}+3c+3)=0$ , with the quadratic factor lacking real roots. We must be careful with the signs of c,s. One solution is $c=-1,s=0$ , other is $c=\frac{12}{,}\text{}s=\frac{\sqrt{3}}{2}$ . In other words, the real x allowed are $x=\pi (2k+1),,x=\frac{\pi (1+6k)}{3}\text{}\text{for}\text{}k\in \mathbb{Z}$

Describe all solutions of Ax=0 in parametric vector form, where A is row equivalent to the given matrix

$$\left[\begin{array}{cccc}1& 3& 0& -4\\ 2& 6& 0& -8\end{array}\right]$$ Find, correct to the nearest degree, the three angles of the triangle with the given vertices

A(1, 0, -1), B(3, -2, 0), C(1, 3, 3)Whether f is a function from Z to R if

?

a) $f\left(n\right)=\pm n$.

b) $f\left(n\right)=\sqrt{{n}^{2}+1}$.

c) $f\left(n\right)=\frac{1}{{n}^{2}-4}$.How to write the expression ${6}^{\frac{3}{2}}$ in radical form?

How to evaluate $\mathrm{sin}\left(\frac{-5\pi}{4}\right)$?

What is the derivative of ${\mathrm{cot}}^{2}x$ ?

How to verify the identity: $\frac{\mathrm{cos}\left(x\right)-\mathrm{cos}\left(y\right)}{\mathrm{sin}\left(x\right)+\mathrm{sin}\left(y\right)}+\frac{\mathrm{sin}\left(x\right)-\mathrm{sin}\left(y\right)}{\mathrm{cos}\left(x\right)+\mathrm{cos}\left(y\right)}=0$?

Find $\mathrm{tan}\left(22.{5}^{\circ}\right)$ using the half-angle formula.

How to find the exact values of $\mathrm{cos}22.5\xb0$ using the half-angle formula?

How to express the complex number in trigonometric form: 5-5i?

The solution set of $\mathrm{tan}\theta =3\mathrm{cot}\theta $ is

How to find the angle between the vector and $x-$axis?

Find the probability of getting 5 Mondays in the month of february in a leap year.

How to find the inflection points for the given function $f\left(x\right)={x}^{3}-3{x}^{2}+6x$?

How do I find the value of sec(3pi/4)?