Express

Question

Express $heta$ in terms of $x$ using the inverse sine, inverse tangent, and inverse secant functions.

Accepted Answer

Hey, everyone. In this problem, we're given ABC, which is a right angled triangle. We have that the hypotenuse has a length of 15. Side AB has a length y. And we're asked to find the measure of angle c in terms of the side y, and to write our answer using arccos, arccotangent, and arccosecant. Now we're given 4 answer choices here, all of them just containing different expressions for that angle c in terms of those three functions. So let's go back up to our diagram and get started on this problem. And the first thing we're gonna do is we're gonna call the side b, c that we're not given information about, we're gonna call that x so that we can use that in our expressions. And we're gonna need that side length, but we're gonna write that side length in terms of y. And we don't want x in our expressions, we want it to only be in terms of y and numbers. So how can we do that? We have a right triangle. We can use Pythagorean Theorem. So we know that x squared plus y squared is going to be equal to the hypotenuse squared, which is 15. So x squared plus y squared is equal to 15 squared. That means that x squared is equal to 225 minus y squared. So we can write x as the square root of 225 minus y squared. And we're just using the positive root here because we're talking about a distance, so we don't have to worry about that negative. Alright. So now we have all three sides written in terms of y. And we have side y itself, we have the hypotenuse, which is a number, and then we have this side x, which is now written in terms of the number and y. So let's start with arccos. Okay. Let's think about the cosine, okay, of angle c. Now cosine of angle c is going to be equal to the adjacent side divided by the hypotenuse. Okay, recall those trig ratios. Now, the adjacent side, well, that's x. But we know x, so we can write that cosine of angle c is going to be equal to the square root of 225 minus y squared divided by the hypotenuse which is 15. Which tells us that angle c is going to be equal to r cosine k taking r cosine on both sides to isolate for c of this entire expression. The square root of 225 minus y squared, all divided by 15. Okay? Alright. So there's our coarse expression. And now we're going to do the exact same thing for our cotangent and our cosecant. Well, let's give ourselves some more room. And we can also eliminate some answer choices while we're at it. So if we look at answer choice b, it has that angle c is going to be cos inverse of y divided by 15. That is not correct, that's not the expression we found, so we can eliminate option b. Okay? That's the incorrect expression for cosine. All right. So let's keep going. Now we're gonna move to cotangent. Now recall that cotangent is equivalent to 1 divided by tangent of C. Sometimes it's easier to just remember the tangent, and then remember that cotangent is 1 divided by tangent, so I want to write both of those. This is gonna be equal to the adjacent side divided by the opposite side. Again, the adjacent side, we can go back to our diagram. We're talking about finding angle c. Okay? So the adjacent side again is x. And now the opposite side is going to be y. Okay. So we're going to end up with cotangent of c equal to the square root of 225 minus y squared divided by y. All right. Now we're going to take our co arc cotangent to isolate for c. We get arc cotangent. Oops. We get that c is equal to arc cotangent of this entire expression we have, which was 225 minus y squared, the square root of all that, divided by y. And that is our expression for arc cotangent. Alright. Back to our answer choices. So we can see that we are going to have option a here. Okay? Option a is the only one that has the correct expression for the arc cosine and arc cotangent, k, or cosinverse, cotangent inverse. Option c has the reciprocal for the cotangent, and so does option d. K? So we're expecting this to be option a, but let's go through into cosecant as well just to double check and be positive. Alright. So for cosecant, what do we have? Cosecant of angle C. Okay, recall that cosecant is gonna be 1 divided by sine of C. Okay, so it's easier to remember sine. Go ahead and do that. And so this is gonna be equal to the hypotenuse divided by the opposite side. Okay? Now the hypotenuse of our triangle is 15. The opposite side to angle c is y, so we get that cosecant of c is just going to be 15 divided by y. We can take arc cosecant, okay, or cosecant inverse, And we are missing a c here. One second. Arc, o secant. There we go. Of our expression which was 15 divided by y. So that is our final expression we were looking for. And looking at our answer choices, we can see that option a is indeed correct. We have the inverse cosine of the square root of 225 minus y squared divided by 15. K. The inverse cotangent square root of 225 minus y squared divided by y, and the inverse cosecant of 15 divided by y. Thanks everyone for watching. I hope this video helped. See you in the next one.

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