Question

Cos 5 x +2 sin 5 x=0

Answer 1

Решение смотрите во вложении

Answer 2

To solve the equation cos(5x) + 2sin(5x) = 0, we can use trigonometric identities to simplify and solve for x. Let's work through it step by step.

First, we'll use the Pythagorean identity: sin^2(x) + cos^2(x) = 1.

Divide the entire equation by cos(5x):

cos(5x) / cos(5x) + 2sin(5x) / cos(5x) = 0 / cos(5x)

1 + 2tan(5x) = 0

Now, subtract 1 from both sides:

2tan(5x) = -1

Finally, divide both sides by 2:

tan(5x) = -1/2

Now, we want to find the values of x that satisfy this equation. Remember that the tangent function has a period of π (180 degrees), so we'll look for solutions within that period.

We need to find the angles whose tangent is -1/2. One solution is in the second quadrant, and another is in the fourth quadrant. These angles can be found using the inverse tangent function (arctan or tan^-1):

Angle in the second quadrant: arctan(-1/2) Angle in the fourth quadrant: π + arctan(-1/2)

Now, let's find the numerical values of these angles:

Angle in the second quadrant: arctan(-1/2) ≈ -26.57 degrees or -π/6 radians Angle in the fourth quadrant: π + arctan(-1/2) ≈ 135.43 degrees or 5π/6 radians

Since the tangent function has a period of π (180 degrees), we can add or subtract integer multiples of π to find all the possible solutions:

5x = -π/6 + kπ 5x = 5π/6 + kπ

Now, solve for x:

x = (-π/6 + kπ) / 5 x = (5π/6 + kπ) / 5

where k is an integer.

These are the general solutions for the equation cos(5x) + 2sin(5x) = 0. They represent all possible values of x that satisfy the given equation.