By "principle angle" I'm assuming you mean "reference angle?" (I've never heard the term before.)

Anyway, to get a reference angle you are looking for the magnitude of the smallest angle between the given angle and either the + or - x axis.

For example, if you look at the angle -187 degrees you will see that it falls in the second quadrant, so the reference angle will be measured from the -x axis. So the reference angle will be 7 degrees.

As to how you would actually go about finding the value of this angle, my first recommendation is to write the angle as a positive angle measured from the +x axis. If the angle is negative, simply add 360 degrees to it. In the case of -187 degrees we have that -187 + 360 = +173 degrees. Then to find the reference angle we simple subtract this from 180 degrees: 180 - 173 = 7 degrees.

Another example:

410

This is bigger than 360 degrees, so subtract 360 degrees:

410 - 360 = 50 degrees.

This is a first quadrant angle, so we are looking for the angle measured from the +x axis. This is just 50 degrees.

A final example:

-67

This is negative so add 360 degrees to it:

-67 + 360 = 293 degrees

This is a fourth quadrant angle, so we are looking for the angle measured from the +x axis. This is 360 - 293 = 67 degrees.

y can't be -7. Cosine only goes from -1 to 1. There's a typo here somewhere.Evaluate y = cos (theta) for 0 degree less than or equal to (Theta) less than or equal to 540 degree when y =-7 (answer to nearest degree)

Use the inverse button on your calculator to find the inverse sine of -0.3. I getEvaluate for when y =-0.3 (answer to nearest degree)

Now, we may add or subtract any multiple of 360 degrees to this. We can't subtract 360 degrees from this because that would be less than -90 degrees. But we can add 360 degrees to this because this is less than 540 degrees. So another solution would be:

As we can't add another 360 degrees to this without it being larger than 540 degrees, we are done. So the solution set is

to the nearest degree.

-Dan