Question

How do you find the cot θ if csc θ = square root of five divided by two and tan θ > 0.

Answer 1

do you want to use Pythagorean identities or do you want to use a drawing.

Answer 2

Pythagorean identities please

Answer 3

well then we know the following:
\[\sin^2(\theta)+\cos^2(\theta)=1 \\ 1+\cot^2(\theta)=\csc^2(\theta) \\ \tan^2(\theta)+1=\sec^2(\theta)\]

Answer 4

which equation I just listed do you think would be most helpful here

Answer 5

the second

Answer 6

exactly
now let's keep in mind this:
if tan is pos then cot is pos
if tan is neg then cot is neg
here we are given that tan>0 which means tan is pos

Answer 7

so we already know what sign the answer will have

Answer 8

now we need to replace csc(theta) with sqrt(5)/2

Answer 9

\[1+\cot^2(\theta)=(\frac{\sqrt{5}}{2})^2 \]

Answer 10

can you simplify the right hand side before we continue

Answer 11

yah

Answer 12

so once you do that
you will solve that equation like you would if you were solving the following for x:
\[1+x^2=(\frac{\sqrt{5}}{2})^2 \\ x^2=(\frac{\sqrt{5}}{2})^2-1 \]
you would then take square root of both sides
and decide which sign to use

Answer 13

so the right side would be 5/4

Answer 14

yah so you have
\[1+\cot^2(\theta)=\frac{5}{4}\]

Answer 15

then how would you continue?

Answer 16

just like I did above

Answer 17

subtract 1 on both sides

Answer 18

then take the square root of both sides
and decide which sign to use

Answer 19

ok one second

Answer 20

\[\cot \theta=\sqrt{5/4-1}\]

Answer 21

right you chose the right sign since cot is positive

Answer 22

now simplify your answer

Answer 23

it would be \[1/2\]

Answer 24

yep yep

Answer 25

Thank you XD