Use the given information to find each of the following.

Question

sin 2x, given sin x = 0.6, π/2 < y < π

Accepted Answer

Welcome back. I am so glad you're here. We are asked to determine the exact value of the S of two Y if the sign of Y equals 0.96 and Y is going to be greater than pi divided by two, but less than pi. Now, our answer choices are answer choice. A negative 271 divided by 576. Answer choice B negative 23 divided by 49 answer choice C negative 336 divided by 625. And answer choice D 451 divided by 576. All right. First, let's take a look at this restriction for our angle. We're told that going to be in between pi divided by two and pi. If we draw a quick sketch of a coordinate plane, we've got a vertical Y axis, a horizontal X axis. They come together at the origin in the middle and that Y axis not to be confused with our angle. Y pi divided by two is the positive part of the Y axis and Pi is the negative part of the X axis. So in between Pi divided by two. And pi we're talking about quadrant two and that'll be important in a little bit. Let's go back and look at the sign of Y equals 0.96. Now, there's actually several different ways to solve this. So if you've gotten to your answer in a different way, that's OK. But one way we can work on it is by changing that 0.96 into a fraction since all of our answers are in fractions, so 0.96 would be 96 divided by 100. And that simplifies to be 24 divided by 25. So if we know that the sign of Y equals 24 divided by 25 we recall from previous lessons that the sign of an angle is equal to the Y value divided by R. Now this Y value is not to be confused with our angle Y. So here we know that our Y equals, we can just look that equals 24 our R value equals 25. And now we're going to need to figure out our X value as well. Why do we need to know our X value? Well, because to get to s of two multiplied by something here, we're using Y that is equal to a double angle identity where the sign of two multiplied by something equals two s of that something Y multiplied by the cosine of that something. Why? And we already know what our sine of Y is. But we don't know what our cosine of Y is. And in order to figure out cosine of Y, we can use the X value because we recall from previous lessons that the cosine of an angle here, we're using Y is equal to X divided by R. And we know our R can put that in the denominator. That's 25. We've just got to figure out the X here. We can look back at our coordinate plane. So we know that our R coming from the origin has a length of 25. We know that our Y value. So that's going up and down in this second quadrant has a length of 24. And we're just trying to find what our X value is here and that is not drawn to scale. So to figure out the X value, we recall from previous lessons that R equals the square root of X squared plus Y squared, let's fill in what we know R is 25 that equals the square root of X squared is unknown plus Y is 24 squared. Bring this down 25 equals the square root of X squared plus 24 squared is 576. Now, we can square both sides. And on the left 25 squared is 625 that equals on the right, that square rut is gone and it's just X squared plus 576 we'll subtract the 576 from both sides on the left, 625 minus 576 is 49. That equals X squared. We'll take the square of both sides. And now we have our X by itself, it equals plus or minus seven. And here we can see that in the second quadrant, our X value is negative, it's to the left of the origin. So we can disregard the positive result. And our X value is just negative seven. We can fill that in, in our numerator for the cosine of Y that's negative 7 25th. Now we just plug in what we know to solve for the sine of two Y, we can bring down the two multiplied by. Now for the sine of Y again, that was 24 25th. And for the cosine of Y that's negative 7 25th and multiplying all of this straight across, we are going to get a negative 336 divided by 625. And that is our sign of two Y. We look at our answer choices and this matches with answer choice. C well done will catch you on the next one.

Use the given information to find each of the following.
sin 2x, given sin x = 0.6, π/2 < y < π

Key Concepts

Double-Angle Identity for Sine

Using the Pythagorean Identity to Find Cosine

Determining the Sign of Trigonometric Functions Based on Quadrants

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