Question

1)sin x/4 = 1 2) cos x/3 = 0 3) cos2x=1 4)tgx/2=0 5)ctg4x=0

Answer 1

1) sinx/4=1   x/4=П/n  x=4*П*n, nЄZ
2) cosx/3=0  x=П/2+П*k     x=3*П/2+3*П*k, kЄZ
3) cos2*x=1 2*x=2*П*n    x=П*n, nЄZ
4) tqx/2=0   x/2=П*k    x=2*П*k, kЄZ
5) ctq4*x=0   4*x=П/2+П*n   x=П/8+П*n/4, nЄZ

Answer 2

Let's solve each of the trigonometric equations one by one:

1. sin(x/4) = 1

To solve this equation, we need to find the values of x that satisfy the equation.

Since sin(x) can only be equal to 1 for certain values of x, we can write:

x/4 = π/2 + 2πn or x/4 = 3π/2 + 2πn

where n is an integer representing the number of complete cycles.

To solve for x, we multiply both sides of the equation by 4:

x = 2π + 8πn or x = 6π + 8πn

where n is an integer.

2. cos(x/3) = 0

Similar to the previous equation, we can write:

x/3 = π/2 + πn or x/3 = 3π/2 + πn

where n is an integer.

Multiplying both sides by 3:

x = (π/2 + πn) * 3 or x = (3π/2 + πn) * 3

Simplifying further:

x = (3π/2) + 3πn or x = (9π/2) + 3πn

where n is an integer.

3. cos(2x) = 1

Since cos(x) can only be equal to 1 for certain values of x, we can write:

2x = 2πn or 2x = 2πn + 2π

Dividing both sides by 2:

x = πn or x = πn + π

where n is an integer.

4. tan(x/2) = 0

To solve this equation, we need to find the values of x that make the tangent of half the angle equal to 0.

tan(x/2) = sin(x/2) / cos(x/2)

For tan(x/2) to be equal to 0, sin(x/2) must be 0.

sin(x/2) = 0

x/2 = πn (where n is an integer)

Multiplying both sides by 2:

x = 2πn (where n is an integer)

5. cot(4x) = 0

To solve this equation, we need to find the values of x that make the cotangent of 4x equal to 0.

cot(4x) = 0 is equivalent to tan(4x) = ∞ (undefined).

Since tan(4x) is undefined when cos(4x) = 0, we need to find the values of x where cos(4x) = 0.

cos(4x) = 0

4x = π/2 + πn or 4x = 3π/2 + πn

Dividing both sides by 4:

x = (π/8) + (π/4)n or x = (3π/8) + (π/4)n

where n is an integer.

These are the solutions to the given trigonometric equations.