Question

√3 sin x - cos x = - 2

Answer 1

(sqrt(3)/2)*sin x - (1/2)*cos x = 1 - разделила на 2 обе части

cos(pi/6)*sin x - sin(pi/6)*cos x = 1 - вместо чисел синусы и косинусы 30 градусов

sin(x-pi/6)=1 - формула sin(a+b)=sin(a)cos(b)+cos(a)sin(b)

x-pi/6 = pi/2 + 2pi*k, k in Integers - очевидно

x = 2pi/3 + 2pik, k in Integers

Answer 2

√3 + sin(x) - cos(x) = -2

To solve for x, we can rearrange the equation to isolate the trigonometric terms:

sin(x) - cos(x) = -2 - √3

Next, we can use the trigonometric identity sin^2(x) + cos^2(x) = 1 to help us solve for x. We can rewrite the equation in terms of sin(x) and cos(x) using this identity:

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

Now, we can substitute y = sin(x) to simplify the equation:

y - 1 + y^2 = -2 - √3 y^2 + y - 1 + 2 + √3 = 0 y^2 + y + √3 - 1 = 0

Using the quadratic formula, we can solve for y:

y = (-b ± √(b^2 - 4ac)) / (2a) = (-(1) ± √((1)^2 - 4(1)(√3 - 1))) / (2(1)) = (-1 ± √(1 - 4√3 + 4)) / 2 = (-1 ± √(5 - 4√3)) / 2

Now that we have the values for y, we can find the corresponding values for x using the inverse sine function:

x = sin^(-1)(y)

So, the solutions for x are:

x = sin^(-1)((-1 + √(5 - 4√3)) / 2) x = sin^(-1)((-1 - √(5 - 4√3)) / 2)

These are the exact solutions for x. We can also use a calculator to find the decimal approximations for these values.