Question

Помогите решить, пожалуйста) 1. Решить тригонометрические уравнения: 1) √2 ѕіn х – 1 = 0 2) tgх/2 - √3 = 0 3) 2соѕ^2 х - 9ѕіnх + 3 = 0 4) 2соѕх + 5ѕіnх = 0 5) соѕ3х – соѕх = 0 6) ѕіn4х = ѕіn2х 7) 3ѕіn2х - 2ѕіnх ∙ соѕх = 1

Answer 1

Ответ:

Объяснение:

1) sinx=1/V2  (V-корень),   x=(-1)^n* p/4+pn  (p=pi)

2) tgx/2=V3,  x/2=p/3+pn,  x=2p/3+2pn

3)2(1-sin^2x)-9sinx+3=0,  2-2sin^2x-9sinx+3=0,

2sin^2x+9sinx-5=0,  D=81+8*5=121,  sinx=(-9+11)/4=1/2,  sinx=(-9-11)/4=-5(не подходит), т.к. |sinx|<=1,   x=(-1)^n*p/6+pn

4) 2cosx+5sinx=0| :cosx не=0,    5tgx+2=0,  tgx=-0,4

x=arctg(-0,4)+pn

Answer 2

1) √2sin(x) - 1 = 0 To solve this equation, we can isolate sin(x) by moving -1 to the other side: √2sin(x) = 1 Now, divide both sides by √2: sin(x) = 1/√2 Since sin(x) = 1/√2 is a special value, we can find the angle that satisfies this condition by using the unit circle or trigonometric identities. In the unit circle, sin(x) = 1/√2 corresponds to π/4 or 45 degrees. Therefore, the solution to this equation is: x = π/4 + 2πn, where n is an integer. 2) tan(x/2) - √3 = 0 To solve this equation, we can isolate tan(x/2) by moving √3 to the other side: tan(x/2) = √3 Now, take the inverse tangent (arctan) of both sides to find the angle: x/2 = arctan(√3) To find the exact value of arctan(√3), we can use the special triangles. In the 30-60-90 triangle, tan(π/3) = √3. So, arctan(√3) = π/3. Multiplying both sides by 2: x = 2(arctan(√3)) Therefore, the solution to this equation is: x = 2(π/3) + 2πn, where n is an integer. 3) 2cos^2(x) - 9sinx + 3 = 0 To solve this equation, we can use the identity cos^2(x) = 1 - sin^2(x): 2(1 - sin^2(x)) - 9sinx + 3 = 0 Expanding and rearranging: 2 - 2sin^2(x) - 9sinx + 3 = 0 -2sin^2(x) - 9sinx + 5 = 0 Now we have a quadratic equation in terms of sinx. To solve this, we can either factor or use the quadratic formula. Using the quadratic formula: sinx = (-b ± √(b^2 - 4ac)) / (2a) a = -2, b = -9, c = 5 sinx = (9 ± √(81 - 4(-2)(5))) / (-4) sinx = (9 ± √(81 + 40)) / (-4) sinx = (9 ± √121) / (-4) sinx = (9 ± 11) / (-4) sinx = -5/2 or 1/2 Since sinx cannot be greater than 1 or less than -1, the solution is sinx = 1/2. Using the unit circle or trigonometric identities, we can find the angles that satisfy sinx = 1/2. The angles that satisfy sinx = 1/2 are π/6 and 5π/6. Therefore, the solutions to this equation are: x = π/6 + 2πn or x = 5π/6 + 2πn, where n is an integer. 4) 2cosx + 5sinx = 0 To solve this equation, we can divide both sides by cosx: 2 + 5tanx = 0 Now, move 2 to the other side: 5tanx = -2 Divide both sides by 5: tanx = -2/5 To find the angle that satisfies this condition, we can use the inverse tangent (arctan) function: x = arctan(-2/5) Using a calculator or reference angles, we find: x ≈ -0.38 radians or -21.8 degrees Therefore, the solution to this equation is: x ≈ -0.38 + πn, where n is an integer. 5) cos3x - cosx = 0 To solve this equation, we can use the identity cos(3x) = cos^3(x) - 3cos(x)sin^2(x): cos^3(x) - 3cos(x)sin^2(x) - cos(x) = 0 Rearranging: cos^3(x) - cos(x) - 3cos(x)sin^2(x) = 0 Now, factor out cos(x): cos(x)(cos^2(x) - 1 - 3sin^2(x)) = 0 cos(x)(cos^2(x) - 1 - 3(1 - cos^2(x))) = 0 cos(x)(cos^2(x) - 1 - 3 + 3cos^2(x)) = 0 cos(x)(4cos^2(x) - 4) = 0 cos(x)(2cos(x) + 2)(2cos(x) - 2) = 0 cos(x)(2cos(x) + 2)(cos(x) - 1) = 0 Now we have three factors to consider: cos(x) = 0, cos(x) + 1 = 0, and cos(x) - 1 = 0. For cos(x) = 0, x can be either π/2 + πn or 3π/2 + πn, where n is an integer. For cos(x) + 1 = 0, x = π, because cos(π) = -1. For cos(x) - 1 = 0, x = 0, because cos(0) = 1. Therefore, the solutions to this equation are: x = π/2 + πn, 3π/2 + πn, π, and 0, where n is an integer. 6) sin(4x) = sin(2x) To solve this equation, we can use the double angle identity for sine: 2sin(2x)cos(2x) = sin(2x) Dividing both sides by sin(2x): 2cos(2x) = 1 Now, divide both sides by 2: cos(2x) = 1/2 To find the angles that satisfy this condition, we can use the inverse cosine (arccos) function: 2x = arccos(1/2) Using reference angles or a calculator, we find: 2x = π/3 or 5π/3 Dividing both sides by 2: x = π/6 or 5π/6 Therefore, the solutions to this equation are: x = π/6 + πn or 5π/6 + πn, where n is an integer. 7) 3sin(2x) - 2sin(x)cos(x) = 1 To solve this equation, we can simplify the equation first: 3sin(2x) - sin(2x) = 1 Expanding: 2sin(2x) = 1 Divide both sides by 2: sin(2x) = 1/2 To find the angles that satisfy this condition, we can use the inverse sine (arcsin) function: 2x = arcsin(1/2) Using reference angles or a calculator, we find: 2x = π/6 or 5π/6 Dividing both sides by 2: x = π/12 or 5π/12 Therefore, the solutions to this equation are: x = π/12 + πn or 5π/12 + πn, where n is an integer.