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: sin 2θ

sin θ = ﹣2/3, θ lies in quadrant III.

Accepted Answer

Hey, everyone in this problem, we're asked to determine the exact value of the expression using the provided data. The expression we're given is sine of two beta. And we're told that sine of beta is equal to negative three quarters with beta lying in quadrant three, we have four answer choices. Option A negative three multiplied by the squared of seven divided by eight. Option B is the same as a just positive option C the square root of seven divided by 24 in option D, the square root of seven divided by eight. And we're gonna start by writing the expression that we're trying to find. So we have sign up to beta. We wanna figure out what this value is. Now, the only information we're given is about sine beta, not anything about sine two beta. So let's use our double ale identities to convert this expression with an angle of two beta into an expression with trig functions with just a single beta. So recall that sign of two beta is equal to two sine beta multiplied by cosine beta. So we can write exactly that we have signed two beta. And so this is gonna be equal to two diana beta multiplied by cosine of beta. Now, the problem tells us sine beta. And so we have this sine beta term. We don't know this coast beta term yet. We don't have this. So what we need to do is find co sign of beta first. OK. So we're gonna find cosine of beta. Now recall that cosine of beta is equal to the X value divided by R. So let's write out Sine beta. That's the information we're given and see if we can relate that to either of these values. So we know that sign of beta is equal to negative three quarters. Recall that sign of beta or sign of theta any angle is gonna be the Y value divided by R Y divided by R. So we can see that in code beta and Sine beta, we both, they both have this R value. OK. So we're gonna be able to relate those and let's think about what why is, and what are is we're in quadrant three. So if we draw just a quick sketch of our coordinate plane, we have Q one in the top right, Q two in the top left, Q three in the bottom, left and Q four in the bottom, right. And we are in Q three if we're in Q three and we can see that we're below the X axis. So the Y value is going to be negative. Hm. So if we're looking at sign of beta which is negative 3/4. That means that Y is going to be negative three, which makes sense why should be negative? Because we're in Q three and R is going to be equal to four. So we have our Y value, we have our R V. We still need our X value in order to determine beta. But recall the relationship between all three of these values. We can use Pythagorean theorem. We have that X squared plus the Y squared is equal to R squared. We know Y we know R this will allow us to calculate X which will give us our value for cosine, which we can then substitute back into our equation for sine of two beta. So we have X squared plus negative three squared is equal to four squared. So X squared plus nine is equal to 16, subtracting nine from both sides. We get that X squared is equal to seven. And finally, we can take the square root, we get that X is equal to plus or minus the square root of seven. And this is really, really important. We don't want to forget that when we take the square root, we get the positive and the negative root, right? And we need to interpret which one makes sense in this problem. OK. Warren quadrant three, we've already talked about how that makes the Y value negative. But that also means the X value is negative because we're to the left of the Y axis. OK. So X should be negative. And so we're gonna take the negative, then we get the X is equal to negative the square root of seven. So if we go back to cosine of beta, we now have our X value we have for our value and we can substitute those in. We get that cosine of beta is equal to negative the square root of seven divided by four. All right. So we've written Coast Beta. Remember what we're actually trying to find, we're actually trying to find an exact value for sine of two beta. So we're gonna go back to that equation. We have sign of two beta is equal to two multiplied by sign of beta multiplied by cosine of beta. This is gonna be equal to two multiplied by sine beta, which we were given negative three quarters multiplied by cosine of beta, which we just calculated negative the squared of seven divided by four. We can simplify, we have a two divided by a four. OK. So we can divide numerator and denominator by two. We're left with negative three halves multiplied by negative the squared of seven divided by four, the negatives will cancel out and we get three multiplied by the square of seven divided by eight. And that is the exact value for this expression that we were looking for. If we 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: sin 2θ
sin θ = ﹣2/3, θ lies in quadrant III.

Key Concepts

Double-Angle Identity for Sine

Determining the Sign of Trigonometric Functions by Quadrant

Using the Pythagorean Identity to Find Cosine

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