In Exercises 15–22, write each expression as the sine, cosine,...

Question

In Exercises 15–22, write each expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. 2 sin 15° cos 15°

Accepted Answer

Hey, everyone in this problem, we're asked to determine the exact value of the following expression by writing it as the sign of a double angle. The expression we're given is two multiplied by sine of 22. degrees multiplied by cosine of 22.5 degrees. We're given four answer choices. Option A one half, option B negative one half, option C negative square root of two divided by two and option D square root of two divided by two. We're gonna start by writing what we're given. So we have two sine of 22. degrees multiplied by cosine of 22.5 degrees. And the question hinted at this and they're saying to use the sign of a double angle. So let's recall our double angle formula. Now, in this case, we have two sign data. The, and we're called through the double angle formula that this is equal to sign up two. Yeah. So in our problem, we're given the left hand side, we want to write it in terms of the right hand side. But for our case, the is gonna be 22.5 degrees. And so two sine of 22.5 degrees multiplied by code sine of 22.5 degrees is gonna be equal to sine of two multiplied by 22.5 degrees. This is equal to sign of 45 degrees. So now we have this in terms of a single trig function, we wanna go ahead and evaluate it. OK. Now we have the angle of 45 degrees. So let's think about one of our special triangles. OK. And you can think about this on the unit circle as well if you prefer to think about it that way. But we're gonna look at our special triangle. So we have our triangle. Let me just make sure that last side straight. There we go, we have a 45 degree angle. Then this is a 90 degree triangle. So we know that if one of these angles is 45 degrees, the other must be 45 degrees as well because all of the angles add up to degrees. And so in our 45 degree triangle, the side lengths will be the same. We're gonna call both side links one. And then according to Pythagorean theorem, the hypotenuse will be the square root of two. OK? Because it'll be the high pot new squared is equal to the first side squared plus the second side squared and one squared plus one squared is two. And then we take the square. So this is our special triangle. We're looking for a sign of this angle and recall that sign is gonna be the opposite side which is one divided by the hypotenuse, which is the square root of two. And so, according to our special triangle, the sign of 45 degrees is equal to one divided by the square root of two. Now we have a square root in the denominator. We wanna go ahead and rationalize this. What that means is make sure that there is no square root in the denominator. So we're gonna multiply both the numerator and denominator by the square root of two. We do that, our fraction remains the same remains equivalent. And if we simplify, we have one multiplied by the square root of two in the numerator that gives us a square root of two. And in the denominator, we have the square of two multiplied by the square of two is equal to just two. OK? And so that we get, we get sorry that our expression is equal to the square root of two divided by two which corresponds with answer choice. D. Thanks everyone for watching. I hope this video helped see you in the next one.

In Exercises 15–22, write each expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. 2 sin 15° cos 15°

Key Concepts

Double-Angle Identities

Evaluating Exact Trigonometric Values

Trigonometric Function Notation and Simplification

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