 In this video we present the solution to question number nine from the practice exam number two for math ten sixty in which case We're given the trigonometric substitution x equals four sine theta And we're trying to rewrite the trigonometric expression sine of two theta over four Using just the algebraic variable x like so so the first thing to do is you want to unwrap this trigonometric substitution here If x equals four sine theta that means sine theta is equal to x over four and using this Observation we can then construct ourselves a right triangle for which we can use this diet this triangle diagram to help us Understand the real the trigonometric ratios associated to angle theta right here because after all if the sine ratio It x over four or sine is opposite over hypotenuse using the Pythagorean Relationship we get that the other side will be 16 minus x squared notice how there's all these 16 minus x squares in the denominator Well, not just in the job there on the numerator should say that's because of that observation right there so now what do we do with this sine of Two theta over four well this triangle is associated to the angle theta We have to translate from two theta to theta and so we're going to do that in this case using the double angle identity for sign So the double angle identity for sign is two sine theta Cosine theta this is still over four so we see that two goes into four two times Like so so we have a factor of one half sitting out in front Sign which we already know is x over four we could reef we could find that out again from the triangle We already knew that cosine of theta we have to take the adjacent side over the hypotenuse like so And so we're going to get the square root of 16 minus x squared over four And so multiplying things together. We have x times the square root of 16 minus x squared We get four times four Which is 16 times 2 is 32 and so we see the correct response is going to equal choice e And I highly recommend all these questions that you do draw a right triangle actually draw it on the page Students who draw these diagrams are going to be much more successful than those who try to work it alone