how many solutions does the equation sin(5x)=1/2 have on the interval [0,2pi]?

ten

sin(5x) = 1/2 has two solutions for 5x, but if you add 2pi,4pi,6pi,8pi to the solutions and then divide by 5, you get 10 solutions for x.

To determine the number of solutions of the equation sin(5x) = 1/2 on the interval [0, 2pi], we need to set up the equation and solve it step by step.

1. Start with the equation: sin(5x) = 1/2

2. Take the inverse sine (arcsin) of both sides to isolate x.

arcsin(sin(5x)) = arcsin(1/2)

3. Apply the arcsin function to both sides of the equation. Note that arcsin has a limited range from -pi/2 to pi/2.

5x = π/6 or 5x = 5π/6

4. Solve for x by dividing both sides of the equation by 5.

x = π/30 or x = π/6

5. Check if the solutions fall within the given interval [0, 2pi].

x = π/30 is within the interval [0, 2π].
x = π/6 is also within the interval [0, 2π].

6. Conclude that there are two solutions: x = π/30 and x = π/6 within the interval [0, 2π].

Therefore, the equation sin(5x) = 1/2 has two solutions on the interval [0, 2π].

To determine the number of solutions of the equation sin(5x) = 1/2 on the interval [0, 2π], we can use the unit circle or trigonometric identities.

First, let's find the solutions of sin(x) = 1/2 on the interval [0, 2π].
The solutions of sin(x) = 1/2 are x = π/6 and x = 5π/6.

Next, let's consider sin(5x) = 1/2.
To find the solutions of this equation, we need to divide the interval [0, 2π] into five equal parts since the coefficient of x is 5.

The first interval is [0, 2π/5], the second is [2π/5, 4π/5], the third is [4π/5, 6π/5], the fourth is [6π/5, 8π/5], and the fifth is [8π/5, 2π].

For each interval, we need to find the solutions of sin(5x) = 1/2.

1. In the interval [0, 2π/5]:
The solutions of sin(5x) = 1/2 can be found by solving 5x = π/6.
Since x must be in the interval [0, 2π/5], the solution is x = π/30.

2. In the interval [2π/5, 4π/5]:
The solutions of sin(5x) = 1/2 can be found by solving 5x = 5π/6.
Since x must be in the interval [2π/5, 4π/5], the solution is x = 5π/30 = π/6.

3. In the interval [4π/5, 6π/5]:
The solutions of sin(5x) = 1/2 can be found by solving 5x = π/6 and 5x = 7π/6.
Since x must be in the interval [4π/5, 6π/5], the solution is x = 7π/30 = π/4.

4. In the interval [6π/5, 8π/5]:
The solutions of sin(5x) = 1/2 can be found by solving 5x = 5π/6 and 5x = 11π/6.
Since x must be in the interval [6π/5, 8π/5], the solution is x = 11π/30.

5. In the interval [8π/5, 10π/5 = 2π]:
The solutions of sin(5x) = 1/2 can be found by solving 5x = 7π/6.
Since x must be in the interval [8π/5, 2π], the solution is x = 13π/30.

Therefore, the equation sin(5x) = 1/2 has 5 solutions on the interval [0, 2π].