Question

2cos^2x-7sinx-7=0 Решите подробно;)

Answer 1

$2cos^{2} x-7sinx-7=0 \\ 2(1-sin^{2} x)-7sinx-7=0 \\ 2-2sin^{2} x-7sinx-7=0 \\ -2sin ^{2} x-7sinx-5=0 /*(-1) \\ 2sin ^{2} x+7sinx+5=0 \\ sinx=t \\2t ^{2}+7t+5=0 \\ D=49-40=9 \\ \sqrt{D} =3 \\ t_{1} = \frac{-7+3}{4} =-1 \\ t _{2} = \frac{7+3}{4} =2.5 \\ sinx=-1 \\ x=- \frac{ \pi }{2} +2 \pi n \\ sinx \neq 2.5$

Answer 2

Solving the Equation 2cos^2x - 7sinx - 7 = 0

To solve the equation 2cos^2x - 7sinx - 7 = 0, we can use various trigonometric identities and techniques. Let's break it down step by step:

Step 1: Simplify the equation The equation can be simplified by using the identity cos^2x = 1 - sin^2x. Substituting this into the equation, we get:

2(1 - sin^2x) - 7sinx - 7 = 0

Simplifying further, we have:

2 - 2sin^2x - 7sinx - 7 = 0

Step 2: Rearrange the equation To solve for sinx, let's rearrange the equation:

-2sin^2x - 7sinx + 2 - 7 = 0

-2sin^2x - 7sinx - 5 = 0

Step 3: Solve the quadratic equation The equation is now in the form of a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula:

The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by:

x = (-b ± √(b^2 - 4ac)) / (2a)

In our case, a = -2, b = -7, and c = -5. Substituting these values into the quadratic formula, we get:

x = (-(-7) ± √((-7)^2 - 4(-2)(-5))) / (2(-2))

Simplifying further:

x = (7 ± √(49 - 40)) / (-4)

x = (7 ± √9) / (-4)

x = (7 ± 3) / (-4)

So, the two possible solutions for x are:

x = (7 + 3) / (-4) = -10 / (-4) = 2.5

x = (7 - 3) / (-4) = 4 / (-4) = -1

Therefore, the equation 2cos^2x - 7sinx - 7 = 0 has two solutions: x = 2.5 and x = -1.

Please note that the solutions are approximate values and may need to be verified.

Summary: The solutions to the equation 2cos^2x - 7sinx - 7 = 0 are x = 2.5 and x = -1.