Find all values of θ, if θ is in the interval [0°, 360°) and has ...

Question

Find all values of θ, if θ is in the interval [0°, 360°) and has the given function value. See Example 6.cos θ = √3              2

Accepted Answer

Hey, everyone in this problem, we're asked to determine all possible values of data in the interval from 0 to 360 degrees which have the same value as the given function value. Now, the function value we're given as sign of data is equal to the square of three divided by two. And we have four possible answer choices. Option A data is 60 degrees, 120 degrees or 300 degrees. Option B data is 60 degrees or 240 degrees. Option C data is 60 degrees or 120 degrees. And option D data is 60 degrees, 120 degrees or 240 degrees. So we're gonna sign up data as the square root of three divided by two. The first thing we wanna do is find a reference angle the prime. Now, in order to do that, we're gonna draw a special triumph. OK? We're called that sin. Theta is equal to the opposite side divided by the hypo. So we want a special triangle where the opposite side of our angle is the square of three. The hypotenuse is two. The good news. Is we have exactly that. OK. So if we have our angle oh high divided by three or 60 degrees, we have squared of three, two for our hippo and one for the other side. And this is one of our special triangles. And so if we have 60 degrees as our angle, then we get squared to three divided by two for sign of that angle. OK. All right. So our reference angle which we'll call theta prime is going to be 60 degrees here. So that means that 60 degrees is a possible solution. However, there could be more. OK. Remember that our reference angle is going to be between zero and 90 degrees. Always, it's always gonna be less than 90 degrees. And so when we're looking at all possible solutions in the interval from 0 to 360 degrees, there may be other options. Now we're told that sine it is positive science data is positive. OK? And if sign of data is positive, then we know our angle must lie in quadrant one or quadrant. OK. So we can go ahead and draw this out. And so in quadruple one, we have this solution where the angle measured from the closest X axis is 60 degrees. And in this case, hey, this is a solution to the original data. One is just six days in quadrant two and we're gonna draw that 60 degrees. And remember that the referencing is always drawn from the nearest x axis. So we're gonna go from the nearest x axis, which is the negative x axis, we're gonna go in a clockwise direction up 60 degrees. And when we want to figure out the value of data, remember that we measure from the positive X axis in the counterclockwise direction. OK. So for this value of the two, we have not 60 degrees, what we have is a 180 degrees which would take us all the way to this negative X axis minus that 60 degrees that we're going backwards. Ok. So theta two is gonna be 180 degrees minus degrees, which gives us a value of 120 degrees. OK? And so we have these two possible solutions beta one and beta two for this problem. OK? And there are no other solutions and the other quadrant sign is going to be negative, which does not match that original function value we were given. And so the correct answer here is gonna be option C data is 60 degrees or degrees. Thanks everyone for watching. I hope this video helped see you in the next one.

Find all values of θ, if θ is in the interval [0°, 360°) and has the given function value. See Example 6.cos θ = √3 2

Key Concepts

Unit Circle

Cosine Function

Inverse Trigonometric Functions

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