SOLUTION: Find the value of sec(theta) if tan(theta)= Negative square root of 114 divided by 19 and sin(theta)>0.

There are two ways to do this problem.  We'll do it both ways:

First way.  θ is in quadrant II because that 
is the only quadrant in which the tangent 
is a negative number and the sine is a 
positive number.

So we draw the angle θ in quadrant II.
The tangent =  = . So
we take the numerator of the tangent  as y, 
(we take it positive because it goes up from the x-axis).
And we take the denominator of the tangent -19 as x, 
(we take it negative because it goes left from the origin).

We use the Pythagorean theorem: 
or r² = x² + y²
   r² = 19² + 
   r² = 361 + 114
   r² = 475
    r =  =  = 

(r is always taken positive).



And since we know that the secant =  = ,

sec(θ) =  = 

--------------------------------

Second way:

Use the identity: 

1 + tan²(θ) = sec²(θ)
1 +  = sec²(θ)
1 +  = sec²(θ)
 +  = sec²(θ)
 = sec²(θ)
 = sec(θ)
 = sec(θ) 
 = sec(θ)
 = sec(θ)

We have to decide which quadrant θ is in just
as we did using the first method.  So the secant
in QII is negative, so the answer is

 = sec(θ)

You can do the problem either way.

Edwin