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The terminal side of angle π in standard position intersects with the unit circle at the point two π, three π, where zero is less than π is less than π over two.
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Find the value of π.
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First, letβs identify the key information in this question.
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Weβre told that angle π is in standard position.
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What does this mean?
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Standard position refers to the position of an angle within a rectangular coordinate system.
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An angle is in standard position if its vertex is at the origin and its initial side lies on the positive part of the π₯-axis.
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We need to consider where the terminal side of this angle is.
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The question tells us that π is between zero and π over two radians, which means the terminal side of this angle must be in the first quadrant.
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Weβre also given some other information about the terminal side of this angle, which is that it intersects with the unit circle at the point two π, three π.
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The unit circle is the circle with center at the origin and a radius of one unit.
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Its equation is π₯ squared plus π¦ squared is equal to one.
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As the terminal side of angle π intersects the unit circle at the point with coordinates two π, three π, then this point is on the circle, which means its coordinates satisfy its equation.
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Substituting two π for π₯ and three π for π¦ into the equation of the unit circle, we have that two π all squared plus three π all squared is equal to one.
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This gives an equation we can solve in order to find the value of π.
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Squaring each of the terms gives four π squared plus nine π squared is equal to one.
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You need to watch out for a common mistake here, which is forgetting to square the two and the three and thinking instead that the equation should be two π squared plus three π squared is equal to one.
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This is an incorrect equation and would lead to an incorrect value of π.
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Returning to the correct equation and simplifying the left-hand side, we have that 13π squared is equal to one.
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Dividing both sides of the equation by 13 gives π squared is equal to one over 13.
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And next, we need to take the square root of both sides of the equation.
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This gives π is equal to plus or minus the square root of one over 13.
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Now the laws of surds tell us that if we take in the square root of a fraction, then this is equal to the square root of the numerator over the square root of the denominator.
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The square root of one is just one.
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So we have π is equal to plus or minus one over the square root of 13.
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Now this would suggest that there are two possible values for π: positive one over root 13 and negative one over root 13.
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But letβs check whether both of these are valid.
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Remember that π is between zero and π over two, which means that itβs situated in the first quadrant.
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This means that both of its coordinates are positive values.
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As two π and three π must, therefore, both be greater than zero, this means we need to take only the positive value of π.
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The value of π is one over the square root of 13.