Question

Sin2x + sinx = 2cosx + 1

Answer 1

2sinx*cos x +sinx-2cos x-1=0
sin x(2 cos x+1)-1(2cos x+1)=0
(2cos x+1)(sinx-1)=0
2cos x+1=0 либо sinx=1
т.е. совокупность 
вроде бы так
x=-+2 pi/3+ 2*pi*n;
x= pi/2+2*pi*n дальше сами

Answer 2

To solve the equation sin(2x) + sin(x) = 2cos(x) + 1, we can use trigonometric identities to simplify it. Let's break it down step by step:

1. Start with the double-angle identity for sine: sin(2x) = 2sin(x)cos(x). Now our equation becomes: 2sin(x)cos(x) + sin(x) = 2cos(x) + 1.

2. Rearrange the terms by subtracting 2cos(x) from both sides: 2sin(x)cos(x) + sin(x) - 2cos(x) = 1.

3. Factor out sin(x) from the first two terms and factor out -1 from the last two terms: sin(x)(2cos(x) + 1) - (2cos(x) + 1) = 1.

4. Now, notice that (2cos(x) + 1) appears as a common factor on both terms: (2cos(x) + 1)(sin(x) - 1) = 1.

5. Divide both sides of the equation by (2cos(x) + 1): sin(x) - 1 = 1 / (2cos(x) + 1).

6. Add 1 to both sides of the equation: sin(x) = 1 / (2cos(x) + 1) + 1.

7. Combine the fractions on the right side: sin(x) = (1 + 2cos(x) + 1) / (2cos(x) + 1).

8. Simplify: sin(x) = (2cos(x) + 2) / (2cos(x) + 1).

Now we have an expression for sin(x) in terms of cos(x). This can be further simplified, but it does not lead to a straightforward algebraic solution for x. To solve for x, we would typically use numerical methods or graphical approaches.