SOLUTION: If sinθ = 6/10, and tanθ less than 0, what is the value of secθ?

If   =   and   is negative, it means that the angle  is in QII.


Hence,  = -  = -  = - .


Then   =  = - .    ANSWER

You can do it the "plug in the formula" way, or you can learn what you're
actually doing.  The following shows you what you're actually doing.  In this
case it's actually easier:

[BTW, don't type "zero" for "theta". "Zero" (0) might have a slanted line
through it on the computer, but "theta" (θ) has a horizontal line through it. To
type θ, type this 
 
and it will come out θ]

Since the sine is 6/10, which is a positive number, that means θ is either in
Quadrant I or Quadrant II.  Since the tangent is less than 0, that means the
tangent is a negative number, and so θ is either in Quadrant II or Quadrant IV.
So we know that θ is in Quadrant II.  So we draw angle θ in its standard
position in Quadrant II with its terminal side swung around counter-clockwise
into the second quadrant, indicated by the red curved line.

 

Next from the end of the terminal side, we draw a perpendicular to the x-axis
(in green), forming a right triangle with one side on the x-axis:

 

Next we are told that the sine of theta is 6/10.  We know that the SINE is
OPPOSITE/HYPOTENUSE, or y/r, so we put the numerator of 6/10, which is 6, y=6,
on the OPPOSITE side of the angle θ (the green vertical side) and we put the
denominator of 6/10, which is 10, r=10, on the HYPOTENUSE of the right triangle,
which is the same as the terminal side of angle θ.

  

Now we calculate the ADJACENT side to θ by the Pythagorean theorem:



Note that we take the value of x as NEGATIVE because it goes LEFT of the origin.
["UP" and "RIGHT" from the origin are positive, "DOWN" and "LEFT" from the
origin are negative.]

So we label the ADJACENT side as x=-8

 

Now that we have drawn that little sketch, we can easily answer the question.

Since we know that the SECANT is the HYPOTENUSE/ADJACENT or r/x, we look at
the picture and see that the HYPOTENUSE is r=10 and the ADJACENT is x=-8,
so we know that secθ = r/x = (10)/(-8) = -10/8 which reduces to -5/4.

Answer = -5/4

Edwin