SOLUTION: Given cot theta =-2 and csc theta < 0, find sin theta and sec theta

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Question 1198723: Given cot theta =-2 and csc theta < 0, find sin theta and sec theta
Found 3 solutions by MathLover1, math_tutor2020, MathTherapy:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

Given:
cot%28theta%29=-2 and csc%28theta%29+%3C+0

find:
+sin+%28theta%29 and sec%28+theta%29
Start with the identity:
1%2Bcot%5E2%28theta%29=csc%5E2%28theta%29
Substitute
cot%5E2%28theta%29=%28-2%29%5E2

1%2B%28-2%29%5E2=csc%5E2%28theta%29
5=csc%5E2%28theta%29
Substitute
csc%5E2%28theta%29=1%2Fsin%5E2%28theta%29
5=1%2Fsin%5E2%28theta%29
sin%5E2%28theta%29=1%2F5
sin%28theta%29=±sqrt%281%2F5%29
sin%28theta%29=±sqrt%285%29%2F5
use negative solution since cot%28theta%29 negative in Q III and Q IV


sec%28+theta%29=1%2Fcos%28theta%29
use cos%5E2%28theta%29=1-+sin%5E2%28theta%29
cos%5E2%28theta%29=1-+1%2F5
cos%5E2%28theta%29=4%2F5
cos%28theta%29=sqrt%284%2F5%29
cos%28theta%29=±%282+sqrt%285%29%29%2F5
then
sec%28+theta%29=1%2F%28%282+sqrt%285%29%29%2F5%29=sqrt%285%29%2F2
or
sec%28+theta%29=1%2F%28-%282+sqrt%285%29%29%2F5%29=-sqrt%285%29%2F2
since sin%28theta%29 and cot%28theta%29 negative in Q III and Q IV

then,
sec%28+theta%29=sqrt%285%29%2F2 in Q IV
and
sec%28+theta%29=-sqrt%285%29%2F2 in Q III


Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

In response to what the tutor @MathLover1 wrote: Cotangent is NOT negative in quadrant Q3.
Take note how sine and cosine are indeed negative here, so,
cot = cos/sin
the two negative items divide to a positive result.

In short, cotangent is positive in Q3.

Since cot(theta) = -2 is negative, this places theta in either Q2 or Q4.

Then we're told csc(theta) < 0, which fully narrows things down to Q4 only.
270 < theta < 360 in degree mode
3pi/2 < theta < 2pi in radian mode
The actual theta value itself doesn't matter; however, this interval helps narrow things down a bit.

Since cot(theta) = -2 = 2/(-1), we could have a triangle with adjacent side 2 and opposite side -1.
I'll make the opposite length "negative" so that we can keep the proper signs in mind. Of course a negative length isn't possible.
It's purely as a means to retain information.
I'm making the 1 negative so that it indicates we're below the x axis, i.e. the y coordinate is negative.

Use the pythagorean theorem to find the hypotenuse is sqrt(5) units long.

This is one way to draw the triangle in Q4

We have:
opposite = -1
adjacent = 2
hypotenuse = sqrt(5)

Then recall that
sin(theta) = opposite/hypotenuse
sec(theta) = hypotenuse/adjacent

I'll let you finish up from here.

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!

Given cot theta =-2 and csc theta < 0, find sin theta and sec theta
I'd advise you not to look at/Ignore that woman's response. 

As cot is negative, theta is either in quadrant 2 or 4. Since csc is also negative (< 0),
theta is either in the 3rd or 4th quadrant. Thus, theta is definitely in the 4th quadrant.