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In this video, we're going to do the following problem.
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If sine of x equals 2 5ths and cosine of x is less than 0, and x is between 0 and 2 pi, we want to determine the exact value of each of the following.
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We want to find cosine of x over 2, sine of x over 2, and tangent of x over 2.
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So this is a classic half -angle formula problem.
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Now, we know sine of x equals 2 5ths.
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We're given that, and cosine is negative.
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So what that's going to do is that's going to force the triangle to be in quadrant 2.
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If cosine is negative, you're either in quadrant 2 or quadrant 3.
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If you're in quadrant 3, that means sine is negative.
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But if you notice in the problem, we're given that sine is essentially positive.
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So that's going to only...
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That's basically going to force us to be in quadrant 2.
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And if you notice, i drew my right triangle accordingly.
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Here's my angle.
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I'll call x.
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And the side opposite to that angle is 2, right? so this is the y value of 2.
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My hypotenuse is 5.
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That's here.
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And by the pythagorean theorem, you should get negative the square root of 21 for the adjacent side.
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It's negative because we're in quadrant.
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We're on the left side of the y -axis, essentially.
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So we know that cosine of x is equal to negative square root of 21 over 5.
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And now we're ready to find everything.
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Cosine of x over 2 is a half...
01:52
There's a half angle identity.
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This is negative the square root of 1 plus cosine x divided by 2...