Question

Sin(2x)+1=cos(x)+2sin(x)tan*( 5x - 2) = -корень 3​ ​

Answer 1

$1)\; \; sin2x+1=cosx+2\, sinx\\\\2\, sinx\cdot cosx-2\, sinx-cosx+1=0\\\\2\, sinx(cosx-1)-(cosx-1)=0\\\\(cosx-1)(2\, sinx-1)=0\\\\a)\; \;cosx-1=0\; ,\; \; cosx=1\; ,\; \; x=2\pi n,\; n\in Z\\\\b)\; \; 2\, sinx-1=0\; ,\; \; sinx=\frac{1}{2}\; ,\; \; x=(-1)^{k}\cdot \frac{\pi}{6}+\pi k\; ,\; k\in Z\\\\Otvet:\; \; x=2\pi n\; ,\; \; x=(-1)^{k}\cdot \frac{\pi}{6}+\pi k\; ,\; n,k\in Z\; .$

$2)\; \; tg(5x-2)=-\sqrt3\\\\5x-2=arctg(-\sqrt3)+\pi n\; ,\; n\in Z\\\\5x-2=-\frac{\pi}{3}+\pi n\; ,\; n\in Z\\\\5x=2-\frac{\pi}{3}+\pi n\; ,\; n\in Z\\\\x=\frac{2}{5}-\frac{\pi}{15}+\frac{\pi n}{5}\; ,\; n\in Z$

Answer 2

To solve the equation sin(2x) + 1 = cos(x) + 2sin(x)tan(5x - 2) = -√3, we'll break it down into smaller steps. First, let's solve sin(2x) + 1 = cos(x):

sin(2x) + 1 = cos(x)

Now, use the identity sin(2x) = 2sin(x)cos(x) and simplify:

2sin(x)cos(x) + 1 = cos(x)

Next, move cos(x) to the left side of the equation:

2sin(x)cos(x) - cos(x) + 1 = 0

Factor out cos(x) from the first two terms:

cos(x)(2sin(x) - 1) + 1 = 0

Now, we have a quadratic equation in cos(x). To solve for cos(x), we set it equal to 0:

cos(x)(2sin(x) - 1) + 1 = 0

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

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

Now, let's move on to the second part of the equation, which is:

2sin(x)tan(5x - 2) = -√3

We can express tan(5x - 2) as sin(5x - 2) / cos(5x - 2):

2sin(x)(sin(5x - 2) / cos(5x - 2)) = -√3

Now, let's clear fractions by multiplying both sides by cos(5x - 2):

2sin(x)sin(5x - 2) = -√3cos(5x - 2)

Now, we have a trigonometric equation with sin and cos terms. To solve this equation, you'll typically need to use trigonometric identities and techniques. Unfortunately, this equation is quite complex, and finding an exact solution might be challenging. You may need to use numerical methods or graphing to approximate the solutions.

Numerical methods involve using a calculator or software to find approximate solutions. Graphing can help you see where the two sides of the equation intersect.

If you have specific values of x that you want to solve for, please provide those, and I can help you find approximate solutions.