Answer:
sec(theta)= [tex]\frac{5\sqrt{21} }{21}[/tex] cot(theta)= [tex]\frac{\sqrt{21} }{-2}[/tex]
Step-by-step explanation:
First step is to figure out what quadrants you are working in so you take the boundary tan theta>0 the only quadrants tan is positive in is 1 and 3.
Cosecant is hypotenuse over opposite so it is impossible to be in quadrant one because all values have to be positive in quadrant one.
Now that you have limited it to quadrant 3, you know that hypotenuse is 5 and the "opposite" is -2 due to hypotenuse not being able to be negative.
Now to solve for Sec theta: Secant is hypotenuse over adjacent so you need to solve your triangle for the "adjacent" value.
You do pythagorean theorum on the triangle [tex]5^{2}[/tex]= [tex]-2^{2}[/tex] + [tex]B^{2}[/tex]
Solve for B, it's [tex]\sqrt{21[/tex]
Now solving for secant again, hypotenuse over adjacent so it would be [tex]\frac{5}{\sqrt{21} }[/tex]
which can be simplified as [tex]\frac{5\sqrt{21} }{21}[/tex]
Solving for Cot: cotangent is adjacent over opposite so it would be [tex]\frac{\sqrt{21} }{-2}[/tex] and that's all.
