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The terminal side of π in standard position intersects with the unit circle at point π΅, with coordinates eight seventeenths, fifteen seventeenths.
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Find sec of π.
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First, letβs sketch out this image.
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Now that we have this rough outline, we can graph point π΅, an eight seventeenths, fifteen seventeenths.
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The terminal side of our π intersects point π΅.
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Because we know that this angle is in standard position, its initial side will be the positive π₯-axis.
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And here is our π.
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Weβre interested in the sec of π.
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We can find the secant ratio by taking the hypotenuse over the adjacent side length.
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To do this though, we need a right angle.
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We can draw a perpendicular line down from point π΅ to create a right angle.
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And because we know how the coordinates work, we know each side length of our triangle.
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The two smallest sides of this triangle are sides eight seventeenths and fifteen seventeenths.
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To solve for secant, we need to know the length of the hypotenuse.
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The Pythagorean theorem can help us solve that.
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Eight seventeenths squared plus fifteen seventeenths squared equals π squared.
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When we add eight seventeenths squared plus fifteen seventeenths squared, we get the whole number one.
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π squared is equal to one.
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And so we take the square root of one.
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The square root of one is one.
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And our hypotenuse here is length one.
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Now if you were paying close attention, you would know that weβre dealing with the unit circle.
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And because point π΅ is located on the unit circle and our vertex is located at point zero, zero, we already know that the hypotenuse would be equal to one, because it is a radius of the unit circle.
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Any angle who has a terminal side on the unit circle and an initial side on the vertex has a hypotenuse of one.
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But if you didnβt remember that fact, you can still use the Pythagorean theorem.
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Okay, back to our problem, we want to know what the secant of this π is.
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Our hypotenuse is one, and the measure of the adjacent side length is eight over 17.
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Sec of π equals one over eight over 17.
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But we need to do some simplification here.
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One divided by eight over 17 is the same thing as one times 17 over eight.
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One times 17 over eight equals 17 over eight.
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And so the secant of our angle equals 17 over eight.