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.

1
cos θ = - —
               2

Accepted Answer

Hey, everyone. Here we are asked to determine all possible values of theta in the interval from zero degrees to degrees, which have the same value as the given function value, which is cosine of theta is equal to negative one divided by the square root of two. Here we have four answer choice options. Answer choice A theta is equal to degrees or 300 degrees. Answer B beta is equal to degrees or 225 degrees. Answer C beta is equal to degrees or 315 degrees. And answer D theta is equal to degrees or 315 degrees. So to begin this problem, we first want to take a look at the given value for cosine of theta. And we want to first notice that this value is negative. So we first need to recall that cosine of theta is equal to X divided by R. And so with this equivalent ratio and our negative given value, we can conclude here that our theta must lie in either the second quadrant or the third quadrant. So now we will have some value for theta in degrees which lies in both the second and the third quadrant. Next, we also need to recall that when cosine of theta is equal to one divided by the square root of two, we know that theta must be 45 degrees. So again, when we have cosine of 45 degrees, the equivalent value is one divided by the square root of two. So now we know we have two possible values of theta which we now can begin solving for. So we will have cosine of 180 degrees again, since we are in either the second or third quadrant, and now we will either add or subtract 45 degrees. So we can begin by just subtracting 45 degrees. And this is equivalent to cosine of 135 degrees, which is also equivalent to negative cosine of 45 degrees. And this value then gives us the given value which is negative one divided by the square root of two. So now we know that one of our values for theta is 135 degrees. So now repeating a similar process, we now need to add 45 degrees to 180. So we have cosine of 180 degrees plus 45 degrees is equal to cosine of 225 degrees. Which is also equivalent to negative cosine of 45 degrees. And we know that this expression is also equivalent to the given value of negative one divided by the square root of two. So now we have found the second value of theta as degrees. So in conclusion, theta is either 135 degrees which leaves us in the second quadrant or we are in the third quadrant at 225 degrees. And with these two values, we are left here with answer choice B where again, theta is equal to 135 degrees or 225 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. 1 cos θ = - — 2

Key Concepts

Cosine Function

Quadrants of the Unit Circle

Inverse Trigonometric Functions

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