Question

sec (59pi /6)

Answer 1

\[\sec(\theta)=\frac{ 1 }{ \cos(\theta) } \implies \sec \left( \frac{ 59 \pi }{ 6 } \right)=\frac{ 1 }{ \cos \left( \frac{ 59 \pi }{ 6 } \right) }\] We are only finding the cosine of the reference angle here, which is pi/6. But we have to find which quadrant it will be in. So we can revolve the angle around the quadrants and we notice that every 12 times you add 30 degrees, you get 360 degrees and you come back to where you started. This implies that adding 30 degrees 60 times would bring us back to where we started which is the 0 degrees mark. But subtracting 30 degrees once, we get 59 additions of 30 and that gives us a reference angle of pi/6 in the fourth quadrant where cosine is positive. So now we just do 1/cos(pi/6) and we know it's going to be positive so we don't have to change the answer.\[\frac{ 1 }{ \cos \left( \frac{ \pi }{ 6 } \right) }=\frac{ 1 }{ \frac{ \sqrt3 }{ 2 } }=\frac{ 2 }{ \sqrt3 }\]
@Maybe