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.

√3
sin θ = - ——
                2

Accepted Answer

Hey, everyone here we are asked to determine all possible values of theta in the interval from zero degrees to 360 degrees which have the same value as the given function value. And here we are given sine of theta is equal to negative one half. Here we have four answer choice options. Answer choice A beta is equal to degrees or 330 degrees. Answer B beta is equal to degrees or 300 degrees. Answer C beta is equal to 300 degrees or 330 degrees. And answer D beta is equal to degrees or 330 degrees. So the first thing we want to recall in this problem is that sign of theta is equivalent to the ratio Y divided by R. And now here our given value for sign of theta is negative and we know from our ratio that we are dealing with Y values when we know that Y is negative in just the third quadrant and the fourth quadrant. So this tells us that we will have two values for data one of which will be in the third quadrant and the other which will lie in the fourth quadrant. And next, we just want to recall that sign of degrees gives us our given value in a positive form as one half. And now we can just utilize these two pieces of information to begin solving for our possible values of theta. So we can begin with the value of theta which will lie in the third quadrant. So that means we will have sine of degrees. And now we need to add 30 degrees as well. And then this gives us sine of degrees. And this is also can be rewritten as negative sign of 30 degrees. And we know that this expression has an equivalent value of the given value which is negative one half. So now we can conclude that one of our values of theta is 210 degrees. Now moving on to our second value, we will now be in the fourth quadrant. So we will have sine of degrees minus 30 degrees. And now simplifying we get sign of degrees and this can be rewritten as negative sign of 30 degrees. And we know again that this expression is equivalent to negative one half. So now we see that our second value four theta is degrees. So just concluding here, we see that theta is equal to either 210 degrees which will bring us to the third quadrant or it is 330 degrees in the fourth quadrant. And with these two values for the, we are left here with answer choice. A where again, theta is equal to 210 degrees or 330 degrees. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

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

Key Concepts

Understanding the Unit Circle

Sine Function and Its Values

Solving Trigonometric Equations

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