For how many values of theta such that 0<theta<360 do we have cos theta = 0.1? (Note that theta is a measure in radians, not degrees!)

I'm kinda confused with the problem? Could someone help me? I'm thinking unit circle, so cos is the x coordinate. It would be easy if theta = degrees, but it is radians, so I'm confused?
IF it were degrees, it would be 2 right.

degrees or radians or grads, it's still 2. What you call the angle does not matter.

If you plan to succeed in trig, get used to thinking in radians. Once you get past the introductory material, most problems will deal with radians.

Thank you sir, I will take your advice.

However, this problem is for an online class, and 2 does not seem to be the answer. Hmm...why?

Wait, is cos x the y coordinate or the x coordinate?

doesn't matter. All of the trig functions are positive in exactly 2 of the 4 quadrants.

Well think about it. The period of cos(x) is 2pi, 114pi = 358.14, so cos(x) repeats 114/2 = 57 times. So the answer is 57*2 = 144, but since cosine is split over y axis on the first period (calculatorsoup dot com/images/trig_plots/graph_cos_pi.gif)

we add 1 to 144, and get 145.

To solve this problem, we can still use the concept of the unit circle. Remember that the cosine of an angle measures the x-coordinate of the corresponding point on the unit circle.

In this case, we are given that cos(theta) = 0.1. We want to find the values of theta between 0 and 360 degrees (or 0 and 2pi radians) for which this equation holds true.

Let's start by finding the reference angle, which is the acute angle formed between the terminal side of theta and the x-axis.

To find the reference angle, you can use the inverse cosine function (also known as arccosine) of 0.1. In mathematical notation, this would be written as:
reference angle = arccos(0.1)

Using a calculator, you can find that the reference angle is approximately 1.4706 radians (or approximately 84.26 degrees).

However, this is just the reference angle. Remember that cosine is positive in the first and fourth quadrants on the unit circle. So, we need to consider the values of theta that are in these quadrants.

To find all the values of theta that satisfy cos(theta) = 0.1, we need to add or subtract the reference angle from 0 and 2pi.

The possible values of theta can be obtained by adding or subtracting the reference angle to the following angles:

1. First Quadrant: theta = reference angle + 0
2. Second Quadrant: theta = pi - reference angle
3. Third Quadrant: theta = pi + reference angle
4. Fourth Quadrant: theta = 2pi - reference angle

Now, let's solve for theta using the given reference angle:

1. For the first quadrant, theta = 0 + reference angle = 1.4706 radians (or approximately 84.26 degrees).

2. For the second quadrant, theta = pi - reference angle = 3.1416 - 1.4706 = 1.6710 radians (or approximately 95.74 degrees).

3. For the third quadrant, theta = pi + reference angle = 3.1416 + 1.4706 = 4.6122 radians (or approximately 264.26 degrees).

4. For the fourth quadrant, theta = 2pi - reference angle = 6.2832 - 1.4706 = 4.8126 radians (or approximately 275.74 degrees).

Therefore, there are four values of theta between 0 and 2pi for which cos(theta) = 0.1.

cos x is positive in QI,QIV.

There's only one angle in each quadrant where cos x = 0.1

Answer key is wrong, if your question is correct.