Помогите пожалуйста с тригонометрией
Решите тригонометрические уравнения:
1)3cos^2-5cosx-8=0
2)8cos^2x-14sinz+1=0
3)5sin^2x+14 sinxcosx+8cos^2x=0
4)2tgx-9ctgx +3=0
5)sin^2x-5cos^2x=2sin2x
6)5cos2x+5=8sin2x-6sin^2x
Ответы
Ответ дал:
0
1) Cosx = t
3t² - 5t -8 = 0
D = 121
t₁ = 16/6 t₂ = -1
Cosx = 16/6 Сosx = -1
нет решений x = π + 2πk , k ∈ Z
2) 8(1 - Sin²x) -14Sinx +1 = 0
8 - 8Sin²x -14Sinx +1 = 0
-8Sin²x -14Sinx +9 = 0
Sinx = t
-8t² -14t +9 = 0
решаем по чётному коэффициенту:
t = (7 +-√(49 +72))/(-8) = (7 +-11)/(-8)
t₁ = 1/2 t₂ =-18/8
Sinx = 1/2 Sinx = -18/8
x = (-1)ⁿπ/6 + nπ, n ∈ Z нет решений.
3)5sin^2x+14 sinxcosx+8cos^2x=0 | : Сos²x ≠ 0
5tg²x + 14tgx +8 = 0
tgx = t
5t² +14t +8 = 0
t = (-7 +-√(49 -40))/5 = (-7 +- 3)/5
t₁ = -2 t₂ = -4/5
tgx = -2 tgx = -4/5
x = -arctg2 + nπ, n ∈ Z x = -arctg 4/5 + πk , k∈Z
4)2tgx-9ctgx +3=0 | * tgx
2tg²x - 9 +3tgx = 0
tgx = t
2t² + 3t -9 = 0
D = 81
t = (-3 +-9)/4
t₁ = -3 t₂ = 6/4 = 1,5
tgx = -3 tgx = 1,5
x = -arctg3 + πk , k ∈ Z x = arctg1,5 + πn , n ∈Z
5) sin^2x-5cos^2x=2sin2x
Sin²x - 5Cos²x - 4SinxCosx = 0 | : Cos²x ≠0
tg²x - 5 - 4tgx = 0
по т. Виета
tgx = 5 или tgx = -1
x = arctg5 + πk , k ∈ Z x = -π/4 + πn , n ∈Z
6) 5cos2x+5=8sin2x-6sin^2x
5( 1 - 2Sin²x) + 5 = 16SinxCosx - 6Sin²x
5 - 10 Sin²x +5 -16SinxCosx +6Sin²x = 0
-4Sin²x - 16SinxCosx +10*1 = 0
-4Sin²x - 16SinxCosx +10(Sin²x + Cos²x) = 0
-4Sin²x -16SinxCosx +10Sin²x +10Cos²x= 0
6Sin²x -16SinxCosx + 10Cos²x = 0
3Sin²x - 8SinxCosx +5Cos²x = 0 | : Cos²x≠0
3tg²x - 8tgx +5 = 0
tgx = (4 +-√1)/3
tgx = 4/3 или tgx = 1
x = arctg4/3 + πk , k ∈ Z x = π/4 + πn , n ∈Z
3t² - 5t -8 = 0
D = 121
t₁ = 16/6 t₂ = -1
Cosx = 16/6 Сosx = -1
нет решений x = π + 2πk , k ∈ Z
2) 8(1 - Sin²x) -14Sinx +1 = 0
8 - 8Sin²x -14Sinx +1 = 0
-8Sin²x -14Sinx +9 = 0
Sinx = t
-8t² -14t +9 = 0
решаем по чётному коэффициенту:
t = (7 +-√(49 +72))/(-8) = (7 +-11)/(-8)
t₁ = 1/2 t₂ =-18/8
Sinx = 1/2 Sinx = -18/8
x = (-1)ⁿπ/6 + nπ, n ∈ Z нет решений.
3)5sin^2x+14 sinxcosx+8cos^2x=0 | : Сos²x ≠ 0
5tg²x + 14tgx +8 = 0
tgx = t
5t² +14t +8 = 0
t = (-7 +-√(49 -40))/5 = (-7 +- 3)/5
t₁ = -2 t₂ = -4/5
tgx = -2 tgx = -4/5
x = -arctg2 + nπ, n ∈ Z x = -arctg 4/5 + πk , k∈Z
4)2tgx-9ctgx +3=0 | * tgx
2tg²x - 9 +3tgx = 0
tgx = t
2t² + 3t -9 = 0
D = 81
t = (-3 +-9)/4
t₁ = -3 t₂ = 6/4 = 1,5
tgx = -3 tgx = 1,5
x = -arctg3 + πk , k ∈ Z x = arctg1,5 + πn , n ∈Z
5) sin^2x-5cos^2x=2sin2x
Sin²x - 5Cos²x - 4SinxCosx = 0 | : Cos²x ≠0
tg²x - 5 - 4tgx = 0
по т. Виета
tgx = 5 или tgx = -1
x = arctg5 + πk , k ∈ Z x = -π/4 + πn , n ∈Z
6) 5cos2x+5=8sin2x-6sin^2x
5( 1 - 2Sin²x) + 5 = 16SinxCosx - 6Sin²x
5 - 10 Sin²x +5 -16SinxCosx +6Sin²x = 0
-4Sin²x - 16SinxCosx +10*1 = 0
-4Sin²x - 16SinxCosx +10(Sin²x + Cos²x) = 0
-4Sin²x -16SinxCosx +10Sin²x +10Cos²x= 0
6Sin²x -16SinxCosx + 10Cos²x = 0
3Sin²x - 8SinxCosx +5Cos²x = 0 | : Cos²x≠0
3tg²x - 8tgx +5 = 0
tgx = (4 +-√1)/3
tgx = 4/3 или tgx = 1
x = arctg4/3 + πk , k ∈ Z x = π/4 + πn , n ∈Z
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