Use the given information to find the exact value of each of t...

Question

Use the given information to find the exact value of each of the following: $tangent 2 theta$

$\sin θ = \frac{12}{13},$

Accepted Answer

Hey, everyone in this problem, we're asked to determine the exact value of the expression tan of two beta using the provided data. And we're told that nine beta is equal to nine divided by 41. And that beta lies in quadrant two, we have four answer choices. Option A negative 1519 divided by 720. Option B is the reciprocal of option A option C is 720 divided by 1519. And option D is the reciprocal of C. So let's start by writing the expression we wanna find which is 10 of two beta and we're looking for the tangent of two beta, but the data we're provided with is sign of beta. And so the information we have is about just this angle beta, not the angle two beta. So let's go ahead use our half angle identities or double angle identities and write tangent of two beta in terms of expressions with just beta in them. And recall that we can write tangent of two beta as two tan beta divided by one minus and squared beta. So in order to write tangent of two beta. What we need to find is tangent of beta first and then we can substitute it into the expression we've written. Now, we're called the tangent of beta is going to be equal to the Y value divided by the X value. All right. So we need to find the Y value. We need to find the X value. Now let's move to what information we're given, we're told that sine of beta is equal to nine divided by 41. Now recall just like tangent is equal to Y divided by X fine is gonna be equal to Y divided by R. Now, we're told that beta is in quadrant two. So let me draw this little Cartesian plane to remind ourselves where quadrant two is in the top right, we have quadrant one in the top left. We have quadrant two, bottom left is quadrant three and the bottom right is quadrant four. So in quadrant two, we're above the X axis which means that the Y values are positive. OK? So we are in quadrant two which tells us the why values are positive. It's always good just to double check that when we're taking the values from the data that we're given. OK. So we can say that Y is going to be equal to nine and R is going to be equal to 41. Now, we don't really need our value right now in our expression for T beta but it is gonna help us find X. All right. So we have our Y value Y is equal to nine. We're gonna be able to substitute that into our equation for tan beta. Now, how can we find X? Well, we know R and we know why. Well, let's recall that Xy and R are related to Pythagorean theory. So we can write that R squared is equal to X squared plus Y squared substituting in the values we know so far 41 squared is equal to X squared plus nine squared. Simplifying 1681 is equal to X squared plus 81. We can subtract 81 from both sides. And we get that X squared is equal to 1600. OK? And finally taking the square root, we get the X is equal to positive or negative 40. OK? And that's really, really important to remember that when you take the square root, you get both the positive and the negative root. And now it's up to us to figure out based on the information we're given which one of those makes sense. So again, we're in quadrant two, that's to the left of the Y axis. So X should be negative. OK. So if X should be negative, we're gonna take that negative work. OK. So again, Q two, it means the X values I'm negative. And so we're gonna take X is equal to negative 40. All right. So we have our X value, we have our Y value. Now we can write tangent of beta, we know that tangent the beta is going to be equal to nine divided by negative 40. And now we can substitute that into our expression for the tangent of two beta. That value we're actually looking for and simplify. So we're just gonna rewrite that expression to remind ourselves since we can't see it on the screen anymore, we have tangent of two beta is equal to two tangent of beta divided by one minus tangent squared of beta. Now substituting in tangent of beta, this is gonna be two multiplied by negative nine divided by 40 divided by one minus nine divided by 40 squared oops. And that should have a negative inside of the nine divided by 40 before it's quick. All right. And now you can see we're done with all of these trig functions we've dealt with the trig aspect of this question. And now this is just an expression with numbers that we need to simplify and we know how to do that. OK. So let's go ahead and start simplifying. So in the numerator we have this two and then we have a divided by 40. OK. So we can divide the numerator and denominator by two. If we divide the numerator by two, that factor of two is gonna go away, dividing the denominator by two is gonna give us 20. And so we're gonna have negative nine divided by 20 in the numerator in the denominator, we're gonna have one minus negative nine divided by 40 squared. And well, the negative squared is just gonna give us a positive value. We're gonna be subtracting 81 divided by 1600. Now, let's find a common denominator in our denominator in order to simplify. So we have negative nine divided by 20 all divided by, we can write one as 1600 divided by 1600. And that we're subtracting 81 divided by 1600. We're gonna move this up into the right hand side so we can keep up the work we've done so far. And we have negative nine divided by 20 all divided by 1519 divided by 1600. Now, when we have a fraction and we're dividing by a fraction, we can multiply by the reciprocal instead. OK? Those are equivalent. So we have negative nine divided by 20 multiplied by 1600 divided by 1519 1600 is divisible by 20. Same with the 20 in the denominator. So we can divide the numerator and denominator by 20 to simplify 1600 divided by 20 gives us 8020 divided by 20 gives us just one in the numerator. We have negative nine multiplied by 80 negative 720. And in the denominator, we have 1519. And this is the exact value of T two beta that we were looking for in this problem. If we go off and compare this to our answer choices, we can see that this corresponds with answer choice. B thanks everyone for watching. I hope this video helped see you in the next one.

Use the given information to find the exact value of each of the following: $tangent 2 theta$
$\sin θ = \frac{12}{13},$

Key Concepts

Trigonometric Ratios and Their Definitions

Sign of Trigonometric Functions in Different Quadrants

Double-Angle Identity for Tangent

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