SOLUTION: Given cos theta = -1/5 and tan theta < 0, find cot theta and sin theta.

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Question 1198721: Given cos theta = -1/5 and tan theta < 0, find cot theta and sin 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:
cos%28+theta%29+=+-1%2F5 and tan+%28theta%29+%3C+0

find:
cot+%28theta%29 and sin+%28theta%29
Start with the identity:
cos%5E2%28+theta%29+=1-sin%5E2%28theta%29
Substitute cos%28+theta%29+=+-1%2F5
-1%2F5=1-sin%5E2%28theta%29
sin%5E2%28theta%29=1-1%2F5
sin%5E2%28theta%29=4%2F5
sin%28theta%29=±sqrt%284%2F5%29
sin%28theta%29=±%282sqrt%285%29%29%2F5
tan+%28theta%29+%3C+0=>sin%28theta%29=-%282sqrt%285%29%29%2F5
then
cos%5E2%28+theta%29+=1-%28-%282sqrt%285%29%29%2F5%29%5E2
cos%5E2%28+theta%29+=1%2F5
cos%28+theta%29 =±sqrt%281%2F5%29
cos%28+theta%29+=±sqrt(5)/5}}}

cot+%28theta%29=cos%28theta%29%2Fsin%28theta%29...........since tan+%28theta%29%3C0+ and sin%28theta%29=-%282sqrt%285%29%29%2F5, use positive solution for cos
cos%28+theta%29+=sqrt%285%29%2F5

so,
cot+%28theta%29=%28sqrt%285%29%2F5%29%2F%28-%282sqrt%285%29%29%2F5%29
cot+%28theta%29=-1%2F2

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

cos(theta) < 0 and tan(theta) < 0 are two given pieces of information.
This narrows theta to quadrant Q2 in the northwest corner.

cos(theta) = adjacent/hypotenuse = -1/5
adjacent = -1
hypotenuse = 5

Use the pythagorean theorem to find the opposite side is 2%2Asqrt%286%29, which is equivalent to sqrt%2824%29, but I'll stick with the first version I mentioned.

Diagram:

adjacent = -1
hypotenuse = 5
opposite = 2%2Asqrt%286%29

The adjacent value is negative since we're in negative x coordinate territory.

Relevant formulas
cot(theta) = adjacent/opposite
sin(theta) = opposite/hypotenuse

I'll let you finish up.

A similar question is found here:
https://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometry-basics.faq.question.1198723.html

Answer by MathTherapy(10555) About Me  (Show Source):
You can put this solution on YOUR website!
Given cos theta = -1/5 and tan theta < 0, find cot theta and sin theta.
As usual, IGNORE everything that that woman states. If you don't, you'll be as confused as you've never been before.
MOST of what she has written is WRONG, and should therefore be DUMPED in the GARBAGE/TOILET!! She even implies that
sin is negative in the 2nd quadrant. How RIDICULOUS and heart-wrenching!!

As cos and tan are negative (< 0), theta is in the 2nd quadrant.

GIVEN: , and tan+%28theta%29+%3C+0) 

        So, 

       

       

       THAT'S ALL there is to this!!