Question

√3cos(π-2,5x)+cos(π/2-2,5x)=0

Answer 1

√3cos(π-2,5x)+cos(π/2-2,5x)=0
по формулам приведения получаем: 
-√3cos2,5x + sin2,5x=0
sin2,5x=√3cos2,5x |:cos2,5x ≠0
tg2,5x=√3
2,5x=arctg√3 + πn, n∈Z 
2,5x= $\frac{ \pi x}{3} + \pi n$ , n∈Z
2,5=5/2
x= $= \frac{ \frac{ \pi }{3} }{ \frac{5}{2}} + \frac{ \pi n} {\frac{5}{2}} = \frac{2 \pi }{15} + \frac{2 \pi n}{5}$ , n∈Z

Answer 2

To solve the equation √3cos(π-2.5x) + cos(π/2-2.5x) = 0, we can simplify it step by step:

1. We notice that the two cosine terms have different arguments. Let's simplify the equation by combining them into a single cosine term. Using the identity cos(π - θ) = -cos(θ), we have: √3cos(π-2.5x) + cos(π/2-2.5x) = 0 √3(-cos(2.5x)) + cos(π/2-2.5x) = 0

2. Now, let's combine the cosine terms into a single expression: -√3cos(2.5x) + cos(π/2-2.5x) = 0

3. We'll use the addition formula for cosine: cos(A - B) = cos(A)cos(B) + sin(A)sin(B). Applying this formula, we get: -√3cos(2.5x) + cos(π/2)cos(2.5x) + sin(π/2)sin(2.5x) = 0 -√3cos(2.5x) + 0cos(2.5x) + 1sin(2.5x) = 0 -√3cos(2.5x) + sin(2.5x) = 0

4. Now, we have a trigonometric equation involving cosine and sine. We can rewrite sine as the square root of 1 minus cosine squared, using the Pythagorean identity sin^2(x) + cos^2(x) = 1: -√3cos(2.5x) + √(1 - cos^2(2.5x)) = 0

5. Let's simplify further by isolating the square root term: √(1 - cos^2(2.5x)) = √3cos(2.5x)

6. Since both sides are non-negative, we can square both sides of the equation: (1 - cos^2(2.5x)) = 3cos^2(2.5x)

7. Expand and rearrange the equation: 1 - cos^2(2.5x) = 3cos^2(2.5x) 1 = 3cos^2(2.5x) + cos^2(2.5x) 1 = 4cos^2(2.5x)

8. Divide both sides by 4: 1/4 = cos^2(2.5x)

9. Take the square root of both sides: ±√(1/4) = ±cos(2.5x) ±1/2 = ±cos(2.5x)

10. Solve for x: cos(2.5x) = ±1/2

To find the values of x that satisfy this equation, we can look for the angles where the cosine function equals ±1/2.

For the positive case, cos(2.5x) = 1/2, we have: 2.5x = arccos(1/2) 2.5x = π/3 + 2πn or 2.5x = 5π/3 + 2πn, where n is an integer. x = (π/3 + 2πn)/2.5 or x = (5π/3 + 2πn)/2.5, where n is