008A Sample Final A, Question 17

Question: Compute the following trig ratios: a) ${\displaystyle \sec {\frac {3\pi }{4}}}$       b) ${\displaystyle \tan {\frac {11\pi }{6}}}$       c) ${\displaystyle \sin(-120)}$

Foundations:
1) How is secant related to either sine or cosine?
2) What quadrant is each angle in? What is the reference angle for each? Answer:
1) ${\displaystyle \sec(x)={\frac {1}{\cos(x)}}}$
2) a) Quadrant 2, b) Quadrant 4, c) Quadrant 3. The reference angles are: ${\displaystyle {\frac {\pi }{4}},{\frac {\pi }{6}}}$, and 60 degrees or ${\displaystyle {\frac {\pi }{3}}}$

Solution:

Since ${\displaystyle \sec(x)={\frac {1}{\cos(x)}}}$, and the angle is in quadrant 2, ${\displaystyle \sec({\frac {3\pi }{4}})={\frac {1}{\cos({\frac {3\pi }{4}})}}={\frac {1}{\frac {-1}{\sqrt {2}}}}=-{\sqrt {2}}}$
The reference angle is ${\displaystyle {\frac {\pi }{6}}}$ and is in the fourth quadrant. So tangent will be negative. Since the angle is 30 degees, using the 30-60-90 right triangle, we can conclude that ${\displaystyle \tan({\frac {11\pi }{6}})=-{\frac {\sqrt {3}}{3}}}$
Sin(-120) = - sin(120). So you can either compute sin(120) or sin(-120) = sin(240). Since the reference angle is 60 degrees, or ${\displaystyle {\frac {\pi }{3}}}$ , So  ${\displaystyle \sin(-120)={\frac {\sqrt {3}}{2}}}$