Question

7. Розв'яжіть рiвняння
3 cos x + 4 sin x = 3.​

Answer 1

3 cos x + 4 sin x = 3
3 • (1-t^2)/(1+t^2) + 4 • (2t)/(1+t^2) = 3
3 - (3t^2)/(1+t^2) + (8t)/(1+t^2)-3 = 0
(3-3t^2 + 8t-3(1+t^2))/(1+t^2) = 0
(3-3t^2+8t-3-3t^2)/(1+t^2) = 0
(-6t^2+8t)/(1+t^2) = 0
-6t^2+8t = 0
-2t(3t-4) = 0
t(3t-4) = 0
t = 0
3t-4 = 0
3t = 4
t = 4/3
——
tg(x/2) = 0
x = 2пk; k ∈ Z
tg(x/2) = 4/3
x = arctg 4/3 + 2пk, k ∈ Z
Ответ : x = 2пk; k ∈ Z