Question

2 cos2x - 7 sin x - 5 =0

Answer 1

Ответ:

х-5=0 cos2=7sin це правильно

Answer 2

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Далее

$-3-4t^2-7t=0\\t=-1\\t=-\frac{3}{4}$

Обратная подстановка

$sin(x)=-1\\x=\frac{3\pi}{2}+2kn,~k\in Z$

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Answer 3

To solve the equation 2cos(2x) - 7sin(x) - 5 = 0, we can use trigonometric identities and techniques to simplify and solve for x.

First, let's make use of the double angle formula for cosine: cos(2x) = 2cos^2(x) - 1.

So, the equation becomes: 2(2cos^2(x) - 1) - 7sin(x) - 5 = 0

Distribute the 2: 4cos^2(x) - 2 - 7sin(x) - 5 = 0

Combine like terms: 4cos^2(x) - 7sin(x) - 7 = 0

Now, we can use the trigonometric identity: sin^2(x) + cos^2(x) = 1.

Rearrange the identity to solve for cos^2(x): cos^2(x) = 1 - sin^2(x).

Substitute this into the equation: 4(1 - sin^2(x)) - 7sin(x) - 7 = 0

Distribute the 4: 4 - 4sin^2(x) - 7sin(x) - 7 = 0

Combine like terms: -4sin^2(x) - 7sin(x) - 3 = 0

Now, we have a quadratic equation in terms of sin(x). Let's solve for sin(x) by factoring or using the quadratic formula:

-4sin^2(x) - 7sin(x) - 3 = 0

Multiply the equation by -1 to make it easier to work with: 4sin^2(x) + 7sin(x) + 3 = 0

Factor the quadratic equation: (4sin(x) + 3)(sin(x) + 1) = 0

Now set each factor equal to zero and solve for sin(x):

1. 4sin(x) + 3 = 0 4sin(x) = -3 sin(x) = -3/4

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

Since the sine function has a range of -1 to 1, there are no real solutions for sin(x) = -1. Therefore, we only consider the first solution:

sin(x) = -3/4

Now, to find the values of x that satisfy this equation, we can use the arcsin (inverse sine) function:

x = arcsin(-3/4)

The arcsin function returns a value between -π/2 and π/2, so we need to consider both positive and negative angles that satisfy the equation:

x = arcsin(-3/4) ≈ -0.8481 radians or approximately -48.58 degrees

Since the sine function is periodic, we can add integer multiples of 2π to the angles to find all possible solutions. Therefore, the general solutions for x are:

x = -0.8481 + 2πn x ≈ -48.58 + 360n

where n is an integer.

So, the equation 2cos(2x) - 7sin(x) - 5 = 0 has infinitely many solutions given by the above expression for x.