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

cos θ = 0.16

I know how to find the answers when it's in fraction form, however, I do not know how to do it when the problem is in decimals. Is it correct to put the decimal into fractions or would that be wrong? Please help!

fractions are just numbers. Why is it any easier to find θ if

cosθ = 4/25

?? You still have to evaluate cos^-1(x) using your calculator. Seems to me that using a decimal is even easier than using fractions.

Now, if you have some problem like

sin(arccos(.16)) then it would seem reasonable to use a fraction, since then it is clear that if

cosθ = 4/25,
sinθ = √84/25

I know how to find the answer with a fraction because those usually come out to become a 30-60-90 or 45-45-90 triangles. Or if they don't I use the pythagorean theorem to find the unknown side and figure out the problem that way.

But I figured it out. The answer was 1.41+2pi(k),4.87+2pi(k).

To solve the equation cos θ = 0.16, it is not necessary to convert the decimal to a fraction. Instead, we can use the inverse cosine function or the use of a calculator to find the solution.

Here's the step-by-step process to solve the equation:

Step 1: Use the inverse cosine function (cos⁻¹) to find the angle whose cosine is 0.16. In other words, we want to find θ such that cos θ = 0.16.

Step 2: If you are using a calculator, look for the "cos⁻¹" or "arccos" function. Enter 0.16 into the calculator and compute the inverse cosine to get the value of θ.

Step 3: Round the value of θ to two decimal places as per the instructions.

Step 4: Repeat this process for different values of k if the question asks for all possible solutions. Since the equation has periodic properties due to cosine function, there will be infinitely many solutions. By adding integer multiples of 2π or 360 degrees to the initial solution(s), you can find more solutions.

Step 5: Write down all the solutions as a comma-separated list, rounded to two decimal places.

For example, let's calculate the initial solution with a calculator:

cos⁻¹(0.16) ≈ 1.405 radians (rounded to two decimal places)

If the problem asks for all possible solutions, you can add multiples of 2π (or 360 degrees) to the initial solution. For instance, the solutions could be:

θ ≈ 1.405 + 2πk , where k is an integer.

In conclusion, you can solve the equation cos θ = 0.16 by using the inverse cosine function (or a calculator) and then adding multiples of 2π to find all the solutions.