Video Transcript
Find the general solution to the equation cot of 𝜋 over two minus 𝜃 is equal to negative one over root three.
We will begin by rewriting the left-hand side of our equation using our knowledge of the cofunction identities. We recall that the tan of 𝜋 over two minus 𝜃 is equal to the cot of 𝜃. And as such, the cot of 𝜋 over two minus 𝜃 is equal to tan 𝜃. This means that tan 𝜃 is equal to negative one over root three. 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.
We notice that the tan of 𝜋 over six radians is equal to one over root three. However, in our equation, we have tan 𝜃 is equal to negative one over root three. As such, we can sketch the graph of 𝑦 equals tan 𝜃 to recall its symmetry. 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. 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. This simplifies to five 𝜋 over six.
Finally, since the tangent function is periodic with a period of 𝜋 radians, we can find the general solution to the equation. 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.
