Question

2sin^2+2sin(x+pi/3)=sinx-1

Answer 1

$2\sin^2x+2\sin(x+ \frac{\pi}{3} )=\sin x-1\\ 2\sin^2x+2(\sin x\cos \frac{\pi}{3}+\cos x\sin\frac{\pi}{3})=\sin x-1\\ 2\sin^2x+2( \frac{1}{2}\sin x+ \frac{ \sqrt{3} }{2}\cos x)=\sin x-1\\ 2\sin^2x+\sin x+\sqrt{3}\cos x=\sin x-1 \\ 2\sin^2x+\sqrt{3}\cos x+1=0 \\ 2-2\cos^2x+\sqrt{3}\cos x+1=0 \\ 2\cos^2x-\sqrt{3}\cos x-3=0$

пусть cos x = t причем |t|≤1, имеем
2t²-√3t-3=0
D=b²-4ac=3+24=27; √D=3√3
t1=(√3-3√3)/4=-√3/2
t2=(√3+3√3)/4=√3 - не  удовлетворяет условие при |t|≤1

Обратная замена
$\cos x=- \frac{\sqrt{3}}{2} \\ x=\pm \frac{5\pi}{6}+2 \pi n,n \in Z$

Answer 2

To solve the equation 2sin^2(x) + 2sin(x + π/3) = sin(x) - 1, we'll start by simplifying the equation and then finding the values of x that satisfy it.

Step 1: Simplify the equation Using the trigonometric identity sin(a + b) = sin(a)cos(b) + cos(a)sin(b), we can rewrite the equation as follows:

2sin^2(x) + 2sin(x)cos(π/3) + 2cos(x)sin(π/3) = sin(x) - 1

Since cos(π/3) = 1/2 and sin(π/3) = √3/2, we can simplify further:

2sin^2(x) + sin(x) + √3cos(x) = sin(x) - 1

Step 2: Rearrange the equation By subtracting sin(x) from both sides of the equation, we get:

2sin^2(x) + √3cos(x) = -1

Step 3: Apply the Pythagorean identity Using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can rewrite the equation:

2(1 - cos^2(x)) + √3cos(x) = -1

Expanding and rearranging, we have:

-2cos^2(x) + √3cos(x) + 3 = 0

Step 4: Solve the quadratic equation Now, we can solve this quadratic equation for cos(x). Let's introduce a variable t for simplicity:

-2t^2 + √3t + 3 = 0

Using the quadratic formula, we have:

t = [-√3 ± √(√3^2 - 4(-2)(3))] / (2(-2)) t = [-√3 ± √(3 + 24)] / (-4) t = [-√3 ± √27] / (-4) t = [-√3 ± 3√3] / (-4) t = (√3(3 ± 1)) / (-4) t = (√3(2) / (-4)) or (√3(-4) / (-4))

Simplifying further:

t = -√3 / 2 or √3

Step 5: Find the corresponding values of x To find the values of x, we need to determine the values of cos(x) that correspond to the values of t we found in Step 4.

For t = -√3 / 2: cos(x) = -√3 / 2

This corresponds to angles x = 5π/6 + 2πk or x = 7π/6 + 2πk, where k is an integer.

For t = √3: cos(x) = √3

This corresponds to angle x = π/6 + 2πk or x = 11π/6 + 2πk, where k is an integer.

Therefore, the solutions to the equation 2sin^2(x) + 2sin(x + π/3) = sin(x) - 1 are:

x = 5π/6 + 2πk, x = 7π/6 + 2πk, x = π/6 + 2πk, x = 11π/6 + 2πk, where k is an integer.