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Find the general solution to the equation cot of 𝜋 over two minus 𝜃 is equal to negative one over root three.
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We will begin by rewriting the left-hand side of our equation using our knowledge of the cofunction identities.
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We recall that the tan of 𝜋 over two minus 𝜃 is equal to the cot of 𝜃.
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And as such, the cot of 𝜋 over two minus 𝜃 is equal to tan 𝜃.
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This means that tan 𝜃 is equal to negative one over root three.
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Next, we recall the special angles zero, 𝜋 over six, 𝜋 over four, 𝜋 over three, and 𝜋 over two radians, together with the values of the tangent of each of these angles.
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We notice that the tan of 𝜋 over six radians is equal to one over root three.
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However, in our equation, we have tan 𝜃 is equal to negative one over root three.
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As such, we can sketch the graph of 𝑦 equals tan 𝜃 to recall its symmetry.
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Drawing horizontal lines at 𝑦 equals one over root three and 𝑦 equals negative one over root three, we can identify solutions that satisfy our equation.
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Using the symmetry of the graph, one solution of tan 𝜃 is equal to negative one over root three is when 𝜃 is equal to 𝜋 minus 𝜋 over six.
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This simplifies to five 𝜋 over six.
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Finally, since the tangent function is periodic with a period of 𝜋 radians, we can find the general solution to the equation.
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The general solution to the equation cot of 𝜋 over two minus 𝜃 is equal to negative one over root three is five 𝜋 over six plus 𝑛𝜋, where 𝑛 is an integer.