Concept Check Suppose that sec θ = (x+4)/x.

Question

Find an expression in x for tan θ.

Accepted Answer

Everyone in this problem, we're asked to determine the value of co tangent of the in terms of X. If Koc cant of the is equal to X plus six divided by X, we're given four answer choices. Option A plus or minus the square root of 12 X plus 36 all divided by X squared. Option B plus or minus the square root of 12 X minus 36 all divided by X. Option C plus or minus the square root of 12 X minus 36 all divided by X squared and option D plus or minus the square root of 12 X plus 36 all divided by X. Now there's a couple ways we can go about this problem. OK. So we can think about CC, can't, we can think about the ratio of sides that we have. And when we're looking at either the hypotenuse the opposite side, the adjacent side work out the missing one and then write cotangent. But a simpler way here is to use our Trigon meric identities. So recall that code tangent and CO C can are related by the following trigonometric identity one plus cotangent squared of theta is equal to KOS squared of theta. And we know what CS can of data is, we were given that the problem so we can substitute that in and then we'll be able to solve for cotangent of data. OK. All right. So what we have is that cotangent squared of theta is going to be equal to cosecant squared of theta minus one and just moving that one from the left hand side to the right hand side and now we'll substitute in our values. So cotangent squared of the is equal to X plus six divided by X all squared minus what? Now let's expand and simplify cotangent squared of theta is going to be equal to. Well, X plus six squared is gonna give us X squared lost 12 X 436 and this is gonna be all divided by X word. So just X squared them. Now we're subtracting one right now, we have a denominator of X squared. So we wanna write our one with that common denominator of X squared so that we can combine these two terms together. So we're gonna multiply the denominator by X squared. So that we have that common denominator. If we do it in the denominator, we need to do it in the numerator as well. And so our minus one is gonna become minus X squared divided by X squared. Now, we have this common denominator, we can combine these into one single fraction. So we have cotangent squared of theta is going to be X squared plus 12 X plus 36 minus X squared all divided by X squared. In the numerator, we have X squared minus X squared. Those two are gonna add to zero. And so we get that co tangent squared of theta is going to be equal to 12 X plus 36 divided by X squared. Now we're gonna go ahead and take the square root on both sides to get our code tangent. We get that code tangent of theta is going to be equal to the square root. And when we take the square root, we get plus or minus. So we have plus or minus the square root of 12 X 436 all divided by the square root of X squared. Now the square root of X squared that's just going to be X or plus or minus X, we have our plus or minus already accounted for. And so we get that code tangent of data, it is going to be equal to plus or minus the square root of 12 X plus 36 divided by X. OK. And that is that final answer. So we were able to use our trigonometric identity relating cotangent and Kant to figure out the value of code tangent given the value of Koan. Let's take a look at our answer choices and compare what we found. And we can see that the correct answer here. Is going to be option D plus or minus the square root of 12 X plus 36 all divided by X. Thanks everyone for watching. I hope this video helped see you in the next one.

Concept Check Suppose that sec θ = (x+4)/x.
Find an expression in x for tan θ.

Key Concepts

Secant Function

Pythagorean Identity

Tangent Function

Watch next