Question

2√3cos2·(3π/2+x/2)=sinx​

Answer 1

$2\sqrt{3} Cos^{2}(\frac{3\pi }{2}+\frac{x}{2})=Sinx\\\\2\sqrt{3}Sin^{2} \frac{x}{2} -Sinx=0\\\\2\sqrt{3}Sin^{2} \frac{x}{2} -2Sin\frac{x}{2}Cos\frac{x}{2}=0\\\\\sqrt{3}Sin^{2} \frac{x}{2} -Sin\frac{x}{2}Cos\frac{x}{2}=0\\\\Sin\frac{x}{2}(\sqrt{3}Sin\frac{x}{2}-Cos\frac{x}{2})=0\\\\1)Sin\frac{x}{2}=0\\\\\frac{x}{2}=\pi n,n\in Z\\\\\boxed{x=2\pi n,n\in Z}\\\\2)\sqrt{3}Sin\frac{x}{2}-Cos\frac{x}{2}=0|:Cos\frac{x}{2}\neq 0\\\\\sqrt{3}tg\frac{x}{2} -1=0$

$tg\frac{x}{2}=\frac{1}{\sqrt{3}}\\\\\frac{x}{2}=arctg\frac{1}{\sqrt{3} }+\pi n,n\in Z\\\\\frac{x}{2}=\frac{\pi }{6}+\pi n,n\in Z\\\\\boxed{x=\frac{\pi }{3}+2\pi n,n\in Z}$

Answer 2

To solve the trigonometric equation 2√3cos^2(3π/2 + x/2) = sin(x), you can follow these steps:

1. Rewrite the trigonometric functions in terms of basic trigonometric identities.

Start by using the identity cos(3π/2 + θ) = -sin(θ) to rewrite the left side of the equation:

2√3(-sin(x/2)) = sin(x)

2. Eliminate the radical on the left side by squaring both sides of the equation:

[2√3(-sin(x/2))]^2 = sin^2(x)

12(sin^2(x/2)) = sin^2(x)

3. Use the double angle identity for sine to express sin^2(x/2) in terms of sin(x):

12(1 - cos(x))/2 = sin^2(x)

6(1 - cos(x)) = sin^2(x)

4. Rearrange the equation to isolate sin(x):

6 - 6cos(x) = sin^2(x)

sin^2(x) + 6cos(x) - 6 = 0

5. Now, you have a quadratic equation in sin(x). You can solve it by substituting u = sin(x) to get:

u^2 + 6u - 6 = 0

6. Solve the quadratic equation for u (sin(x)) using the quadratic formula:

u = [-b ± √(b^2 - 4ac)] / (2a)

In this case, a = 1, b = 6, and c = -6:

u = [-6 ± √(6^2 - 4(1)(-6))] / (2(1))

u = [-6 ± √(36 + 24)] / 2

u = [-6 ± √60] / 2

u = [-6 ± 2√15] / 2

u = -3 ± √15

So, sin(x) can have two values:

sin(x) = -3 + √15 sin(x) = -3 - √15

7. Now, you can find the values of x by taking the inverse sine (arcsin) of these values:

For sin(x) = -3 + √15:

x = arcsin(-3 + √15)

For sin(x) = -3 - √15:

x = arcsin(-3 - √15)

Note that the solutions will depend on the domain of x you are interested in, typically between 0 and 2π or -π and π for periodic solutions.