How do I find the remaining functions?

Question

I'm having trouble understanding the question.

 

The question is Sec theta=-4, sin theta<0, find the remaining functions. 

Answer 1

Since it's given that Sec theta=-4, sin theta<0, by taking reciprocal of the secant, we find that cos θ = -¼.

The fact that both sine and cosine are negative puts us in the third quadrant.

By the Pythagorean identity, sin θ must = -√(15)/4. 

Now you can easily find tangent, cotangent and cosecant.

Answer 2

Start by drawing a right triangle in standard position in the correct quadrant according to the given conditions.  Hint there is only one quadrant where the sine and secant are both negative.  Next use the definition that the sec(theta) = Hyp/Adj.  We can write any number into a fraction by putting it over 1.  So we have sec(theta) = -4/1 or 4/-1.  It should be obvious what fraction to use and what value goes with the hypotenuse and what value goes with the adjacent.   Now label the sides of the triangle.  Use Pythagorean's Theorem to find the missing side.  Remember to use the correct sign.  Now find the remain trig functions by using the definitions.

 

Hint.  Definitions of trig functions

 

1) sin(theta) = O/H

2) cos(theta) = A/H

3) tan(theta) = O/A

4)csc(theta) = H/O

5) sec(theta) = H/A

6) cot(theta) = A/O