Question

|sinx| = sinx + 2*cosx interval (0.3π) сумма всех решений

Answer 1

1) sinx ≥ 0
sinx = sinx + 2cosx
2cosx = 0
cosx = 0
x = π/2 + πn
Учитывая, что sinx ≥ 0,
x = π/2 + 2πn
Найдем корни, принадлежащие интервалу:
0 < π/2 + 2πn < 3π
- π/2 < 2πn < 5π/2
- 1/4 < n < 5/4
Так  как n - целое,
n = 0      x = π/2
n = 1      x = 5π/2

2) sinx < 0
- sinx = sinx + 2cosx
2sinx + 2cosx = 0
sinx + cosx = 0
tgx + 1 = 0
tgx = - 1
x = - π/4 + πk
Учитывая, что sinx < 0,
x = - π/4 + 2πk
Найдем корни, принадлежащие интервалу:
0 < - π/4 + 2πk < 3π
π/4 < 2πk < 13π/4
1/8 < k < 13/8
Так  как k - целое,
k = 1      x = 7π/4

Сумма корней, принадлежащих промежутку:
π/2 + 5π/2 + 7π/4 = 3π + 7π/4 = 19π/4

Answer 2

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Answer 3

To solve the equation |sinx| = sinx + 2*cosx in the interval (0, 3π/2), we need to consider two cases:

Case 1: sinx ≥ 0 In this case, the absolute value can be removed, so we have sinx = sinx + 2cosx, which simplifies to 2cosx = 0. This equation is satisfied when x = π/2 and x = 3π/2.

Case 2: sinx < 0 In this case, the absolute value becomes negative, so we have -sinx = sinx + 2cosx, which simplifies to 3sinx = -2*cosx. Dividing both sides by cosx (which is negative in this case) gives us:

3*tanx = -2

Solving for x in the interval (0, 3π/2) gives us:

x = arctan(-2/3) ≈ 0.9828 rad and x = arctan(-2/3) + π ≈ 2.124 rad

Therefore, the sum of all solutions is:

π/2 + 3π/2 + arctan(-2/3) + arctan(-2/3) + π = 2π + 2*arctan(-2/3) ≈ 5.281 radians