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If π exists between 180 degrees and 360 degrees and sin π plus cos π equals negative one, find the value of π.
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Letβs first consider the equation sin π plus cos π equals negative one.
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Our first step here is to square both sides of the equation.
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This gives us sin π plus cos π all squared is equal to negative one squared.
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In order to square sin π plus cos π, we need to write the brackets or parentheses out twice.
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Negative one squared is equal to one, as multiplying a negative number by another negative gives us a positive.
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We can expand or distribute the parentheses using the FOIL method.
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Multiplying the first terms, sin π and sin π, gives us sin squared π.
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Multiplying the outside terms gives us sin π cos π.
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Multiplying the inside terms also gives us sin π cos π.
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And finally, multiplying the last terms gives us cos squared π.
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This leaves us with the equation sin squared π plus sin π cos π plus sin π cos π plus cos squared π is equal to one.
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The middle two terms are the same, so they can be grouped or collected.
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This leaves us with two sin π cos π.
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We can also drop down the other three terms, sin squared π, cos squared π, and one.
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One of our trigonometrical identities states that sin squared π plus cos squared π is equal to one.
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This means that the equation could be rewritten as two sin π cos π plus one equals one.
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Another one of our identities, one of the double-angle formulae, states that two sin π cos π is equal to sin two π.
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This means that sin two π plus one is equal to one.
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We can subtract one from both sides of this equation.
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This means that sin two π is equal to zero.
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We need to solve this equation for all values of π between 180 degrees and 360 degrees, exclusive.
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Not including 180 or 360.
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If we consider the graph of sin two π, it has a maximum value of one and a minimum value of negative one.
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The values of π that give us these maximum and minimum values are shown on the graph.
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Sketching this graph shows us that there are numerous values where sin two π is equal to zero.
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They occur at zero degrees, 90 degrees, 180 degrees, 270 degrees, and 360 degrees.
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However, since weβre looking for values between 180 and 360 degrees, the only solution in this case is π equals 270 degrees.
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If sin π plus cos π is equal to negative one, the solution between 180 and 360 degrees is 270 degrees.