Find all solutions of the given equation. (Enter your answers as a comma-separated list. Let k be any integer. Round terms to two decimal places where appropriate.)

2 cos θ − 1 = 0

To find all the solutions of the equation 2 cos θ − 1 = 0, we can start by isolating the cosine term.

Add 1 to both sides of the equation:
2 cos θ = 1

Next, divide both sides of the equation by 2:
cos θ = 1/2

Now, we need to find all the angles θ that have a cosine of 1/2. To do this, we can use the unit circle or refer to the inverse cosine function.

The unit circle shows us that the cosine of π/3 and 2π/3 is equal to 1/2.

However, since we are asked to provide a general solution, we can use the inverse cosine function to find the solutions for any k, where k is any integer.

Using the inverse cosine function, we have:
θ = cos^(-1)(1/2)

Using a calculator, we can find the value of θ in radians:
θ = 1.0472 radians

Since cosine is a periodic function with a period of 2π, we can add multiples of 2π to the solution to find all solutions.

θ = 1.0472 + 2πk, where k is any integer

Rounding to two decimal places, the solutions are approximately:
θ ≈ 1.05 + 6.28k, where k is any integer

Therefore, the solutions of the equation 2 cos θ − 1 = 0 are given by:
θ ≈ 1.05 + 6.28k, where k is any integer

To find all solutions of the given equation 2 cos θ - 1 = 0, we can start by isolating the cosine term.

1. Add 1 to both sides of the equation:
2 cos θ = 1

2. Divide both sides of the equation by 2:
cos θ = 1/2

Now, we need to find all the possible values of θ that satisfy this equation. Remember that the cosine function has a period of 2π.

To find the main solution within the interval [0, 2π), we can take the inverse cosine (or arccos) of 1/2. The inverse cosine function will return the angle whose cosine is equal to 1/2.

3. Take the inverse cosine of both sides:
θ = arccos(1/2)

The inverse cosine of 1/2 is π/3 or 60 degrees. However, since we need the solution in radians and rounded to two decimal places, we'll use the radian measure.

4. Convert the angle to radians and round to two decimal places:
θ ≈ 1.05 radians

Now, we need to find the other solutions within the interval [0, 2π). Since the cosine function has a period of 2π, we can generate additional solutions by adding or subtracting multiples of 2π from the main solution.

5. Add or subtract multiples of 2π from the main solution:
θ ≈ 1.05 + 2πk radians,
where k is an integer.

These values of θ represent all the solutions of the equation 2 cos θ - 1 = 0 within the interval [0, 2π). To obtain a comma-separated list of solutions, substitute different integer values for k and round the results to two decimal places:

θ ≈ 1.05, 7.18, 13.20, 19.25, ...

solve for cosθ the way you would for any other variable.

2cosθ = 1
cos θ = 1/2

ta da. Of course, now you have to now how to find θ, but that's no problem, right?

Find the real solutions of the equation. (Round your answers to three decimal places. Enter your answers as a comma-separated list.)

3.4x4 − 1.9x2 = 2.2