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Find the slope of the tangent line to the polar curve π equals cos two π at the point where π equals π by six.
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Remember, to find the slope of the tangent line to a curve at a given point, we need to begin by calculating the derivative.
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For a polar curve of the form π equals π of π, itβs some function of π, dπ¦ by dπ₯ is equal to dπ¦ by dπ divided by dπ₯ by dπ.
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We use the formula dπ¦ by dπ equals dπ by dπ times sin π plus π cos π and dπ₯ by dπ equals dπ by dπ times cos π minus π sin π.
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Now, if we go back to our question, we see that π is defined as cos of two π.
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And since our equations for dπ¦ by dπ and dπ₯ by dπ are in terms of dπ by dπ and π, weβre going to need to begin by calculating dπ by dπ.
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To do so, we quote the general result that the derivative of cos of ππ₯ is equal to negative π sin of ππ₯.
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And thatβs great, because that means dπ by dπ is negative two sin of two π.
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Letβs now work out dπ¦ by dπ and dπ₯ by dπ. dπ¦ by dπ is dπ by dπ times sin π.
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So, thatβs negative two sin two π sin π plus π times cos π.
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And in this case, thatβs cos two π times cos π.
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We then use the formula for dπ₯ by dπ.
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Itβs dπ dπ times cos π.
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So, thatβs negative two sin two π times cos π.
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We then subtract π times sin π.
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So, here, we subtract cos two π sin π. dπ¦ by dπ₯ is the quotient of these.
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Itβs dπ¦ by dπ divided by dπ₯ by dπ.
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So, itβs negative two sin two π sin π plus cos two π cos π all over negative two sin two π cos π minus cos two π sin π.
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Now, remember, weβre looking to work out the slope of the tangent line to our curve at the point where π is equal to π by six.
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Weβll achieve this by substituting π equals π by six into our expression for the derivative.
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And this gives us the expression shown.
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When we evaluate the numerator, we get negative root three over two plus root three over four.
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And the denominator, we get negative three over two minus one-quarter.
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The quotient of these is root three over seven.
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And so, we see the slope of the tangent line to the polar curve π equals cos two π at the point where π is equal to π by six is root three over seven.