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Find sec π, given π is in standard position and its terminal side passes through the point four-fifths, three-fifths.
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To answer this question, itβs helpful to recall the unit circle.
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Remember, this circle has a radius of one, and we can add the following values of π to our graph by moving in an anticlockwise direction.
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We start at zero, then π over two, π radians, three π over two radians, and finally the full turn takes us back to the start, or two π radians.
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The terminal side is the side that determines the angle.
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In the case of the unit circle, itβs the radius.
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Since we know the terminal side passes through the point four-fifths, three-fifths, we can add π to our diagram.
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The ordered pair four-fifths, three-fifths lies in the first quadrant.
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So, the value of π is between zero and π over two radians.
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Now this ordered pair actually tells us the dimensions of our right-angled triangle.
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Since the π₯-value is four-fifths, the length of the side adjacent to the angle π is four-fifths.
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The π¦-value in our ordered pair is three-fifths, so the length of the side opposite to the angle π is three-fifths.
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Remember, itβs a unit circle.
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The radius of this circle is one, so the length of the hypotenuse is also one.
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This helps us hugely since we can work out the values of sin, cos, and tan π using standard right angle trigonometry.
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But how can we find the value of sec π?
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Well, sec π is one over cos π, so weβll first find the value of cos π.
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Cos π is equal to adjacent over hypotenuse.
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We can label our right-angled triangle relative to the angle π to get the opposite is three-fifths, the adjacent is four-fifths, and the hypotenuse is one.
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Substituting these values into the formula for cos π gives us cos π is equal to four-fifths divided by one, which is simply four-fifths.
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Since sec π is one over cos π, for our value of π, sec π is one over four-fifths, or one divided by four-fifths.
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To divide by a fraction, we change the divide sign to a multiply and find the reciprocal of the second fraction.
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Thatβs one multiplied by five over four, which is simply five over four.
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Sec π then is five over four.