Question

Find the values of the six trigonometric functions of θ.

Function Value: csc θ= 4
Constraint: cot θ < 0

Someone explain this step by step please!

Answer 1

there is a picture of an angle whose sine is \(\frac{1}{4}\) and whose cosecant is \(4\)
all you need is the other side, which you get via pythagoras
then use the ratios to find the other trig function, keeping in mind that you are in quadrant II since sine is positive, and cosine is negative

Answer 2

so r= 4? sin = 1/4?

Answer 3

\[\csc(\theta)=4\iff \sin(\theta)=\frac{1}{4}\]

Answer 4

Yeah I know, so you're saying the hypotenuse is 4, right?

Answer 5

yeah i am saying the hypotenuse is 4 and the "opposite" side is 1
this is just a crutch, but it makes it easy to find the other side and the other ratios that you need

Answer 6

Alrighty so since the hypotenuse (r) is 4 and sin (y) is 1 that means i need to find cos (x), right?

Answer 7

I still do not understand...

Answer 8

you need to find the other leg of the triangle

Answer 9

For the other leg of the triangle i got √(15)

Answer 10

you are missing something in your thinking
there is no \(y\) here, you do not have \(sin(y)=1\)

Answer 11

what you have is
\[\sin(\theta)=\frac{1}{4}\] which you know because you are told
\[\csc(\theta)=4\]

Answer 12

yes, the other sides is \(\sqrt{15}\)

Answer 13

now using "adjacent over hypotenuse" for cosine, you see that
\[\cos(\theta)=-\frac{\sqrt{15}}{4}\]

Answer 14

oh okay, but why is it negative?

Answer 15

good question

Answer 16

it could be either \(\frac{\sqrt{15}}{4}\) or \(-\frac{\sqrt{15}}{4}\)

Answer 17

you know it is the second one because you are told two things at the beginning
cosecant is positive, and cotangent is negative
the first tells you sine is positive as it is the reciprocal
the second tells you cosine must be negative because sine is positive

Answer 18

Oh alright, so would tan (θ)= -√(15)

Answer 19

I mean √(15)/15?

Answer 20

please, i need to know

Answer 21

yes except it is negative

Answer 22

cot θ < 0 is what you are told
since cotangent is negative, so is tangent

Answer 23

Alright i fully understand this question now. Thank you very much!