Video Transcript
Find 𝜃 in degrees given sec of 180
degrees plus 𝜃 is equal to negative two root three over three where 𝜃 is the
smallest positive angle.
In order to solve this problem, we
recall that sec 𝜃 is equal to one over cos 𝜃. It is the reciprocal of cos 𝜃. We can therefore rewrite the
equation as one over cos of 180 plus 𝜃 is equal to negative two root three divided
by three. If two fractions 𝑎 over 𝑏 and 𝑐
over 𝑑 are equal, then their reciprocals must also be equal. This means that in our question,
cos of 180 degrees plus 𝜃 is equal to negative three over two root three. Rationalizing the denominator of
the right-hand side by multiplying the top and bottom by root three gives us
negative root three over two. This is equal to cos of 180 degrees
plus 𝜃.
We recall that cos of one of our
special angles, 30 degrees, is equal to root three over two. Using our CAST diagram, we can find
the angles where cos of 𝜃 is equal to negative root three over two. These will be equal to 180 minus 𝜃
and 180 plus 𝜃. cos of 150 degrees and cos of 210 degrees are equal to negative
root three over two. This means that 180 plus 𝜃 is
equal to 150, and 180 plus 𝜃 is equal to 210. Subtracting 180 from both sides of
the equations gives us 𝜃 equals negative 30 and 𝜃 equals 30.
We are told in the question that 𝜃
must be a positive angle. The smallest positive solution of
sec of 180 degrees plus 𝜃 is equal to negative two root three over three is 30
degrees.
