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: $sine 2 theta$

$\sin θ = \frac{12}{13},$

Accepted Answer

Hello, everyone. We are asked to determine the exact value of the expression using the provided data. The expression we want the value of is the sign of two beta. When the sign of beta equals 9 40 firsts and beta lies in quadrant two, we are given four answer choices. The first thing I want to do is I want to find what the double angle identity is for the sign of two beta. And I recall that that equals two multiplied by the sign of beta multiplied by the cosine of beta from there. I know that the sign of beta is nine 41st. And I recall that when I think about sign and cosine ratios in terms of a coordinate plane that this is equivalent to the ratio of Y over R and that the cosine of beta will be equivalent to the ratio of X divided by R. Right now. I don't know a value for X X is my unknown for this moment. I do know that Y equals nine and R equals 41. So in order to solve my double angle identity, I need to find the value of the cosine A beta. And to do that, I need to first find the value of X. What's helpful is this is based on the Pythagorean theorem. So we can work with X squared plus Y squared equals R squared plugging in the values we know and solve for the X value. So X squared plus nine squared equals 41 squared. So X squared plus 81 equals 41 squared, which is 1681. I wanna get the X variable by itself. So I'm subtracting 81 from both sides, X squared equals 1600. Recall that the opposite of a square is the square root. So taking the square root of both sides, I get that X equals plus or minus 40. So how do I know if it's plus or minus? Well, we are told that beta lies in quadrant two and in quadrant two, our X values are negative. So our X in this case will also be negative. So X will equal negative 40. Now that I know the value for X, I can say that the cosine of beta is a ratio of negative 40 two R which is having both. Now sign and cosign of beta. I'm gonna go back to my double angle identity. So we're the sign of two beta equals two multiplied by the sign of beta. So the sign of beta was nine divided by 41 multiplied by the cosine of beta. Which we just discovered as negative 40 divided by 41. And now I'm gonna do the multiplication. And when I multiply across the numerators, I get negative 720 and multiplied by 41 is 1681. So we now know that the exact value for the sign of two beta is negative 720 divided by 1681 which is answer choice D have a nice day.

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

Key Concepts

Understanding the Quadrants and Sign of Trigonometric Functions

Double-Angle Identity for Sine

Using the Pythagorean Identity to Find Cosine

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