Question

Решить уравнение:

2tgx+2ctgx=5

Answer 1

ctg x=1/tg x, tgx=t:

2t+2/t=5

2t^2-5t+2=0

D=25-4*2*2=9

t=(5+-3)/4

t=2 or t=1/2

 

tgx=2 or tgx=1/2

x=arctg2+pi*k, k in Integers OR x=arctg0.5 + pi*m, m in Integers

Answer 2

2tgx + 2ctgx = 5

2tgx + 2/tgx - 5 = 0

2tg²x - 5tgx + 2 = 0

замена: tgx = a

2a² - 5a + 2 = 0

a₁ = 2, a₂ = 1/2

обратная замена:

tgx = 2

x = arctg2 + πn, n∈Z

tgx = 1/2

x = arctg(1/2) + πn, n∈Z