In Exercises 1–6, use the figures to find the exact value of each trigonometric function.

tan 2θ

Video transcript

Hello, everyone. We are asked to determine the exact value of the trigonometric function. Using the provided figure, we are given a right triangle with a hypotenuse measuring 53. The leg opposite angle beta measures 28 the leg adjacent to angle beta measures 45. We are asked to find the value of the tangent of two beta. We are given four answer choices first. Before we do anything else, you want to use your double angle identities and recognize that the tangent of two beta will equal two times the tangent of beta divided by one minus the tangent squared of beta. So the first thing we should do here is find out what is the tangent of beta. So tangent of beta recall that as a ratio, a tangent is the opposite leg divided by the adjacent leg. So here the like opposite beta has a value of and the leg adjacent to beta has a value of 45. So the tangent of beta is 28 45th. So now we're gonna plug this in to that identity and see what math we can do. So in my new reader, I get two multiplied by 45th. And then in my denominator, I have one minus 28 45th squared. So simplifying my numerator, I'll multiply straight across the top of that fraction and I will have 56 45th in my denominator, the one minus will stay put. And I need to find out what 28 45th squared is the 28 45th squared gives us 784 divided by 2025. So I'm gonna leave the numerator as is so 45th and in my denominator, I need to do one minus divided by 2025. And when I do that, I will have over 2025. I want to clean up my complex fraction so I can rewrite this as 56 45th divided by 1241 over 2025. Remembering my rules of fractions, multiplication of the reciprocal is the same as division. So I'll have 56 divided by 45 multiplied by 2025 divided by 1400. I'm sorry, 1241. I'm going to cross reduce what I can to try to make these numbers a little simpler. Um I know five goes into both 2025. So let's see how that helps me. All right, let's see, does that get any simpler? Um Will nine go into both of them? I think I can reduce this again because nine goes into nine once and goes into 405. Uh It looks like 45 times and I'm not sure anything simplifies across the t the other diagonal. And so when I multiply straight across, I get divided by 1241. And that is gonna be what? The tangent of two beta equals 2520 divided by 1241. And that matches answer choice D have a nice day.