Question about why the signs are wrong in this trig problem

[land_of_ice]

Homework Statement

The problem is: cosθ=-4/5 θ in quadrant 3
and they ask that you find the exact value of the remaining trig functions,

Homework Equations

I do not know all of the relevant things to do, which is why I have a question, about what they are basically, as far as quadrant is concerned, the sign of the answer and whether or not it matters that a picture is drawn and used or not.

The Attempt at a Solution

I got the exact opposite signs to my answer, as the book got, but the values were correct, also I don't know why but after doing about 15 of these problems, when I drew the picture (first step), the θ seemed to be in the same area of the unit circle every time , in quadrant 1, so how can you tell where θ is supposed to be? Also, after I drew the picture, I never use it in the equation, I just plug in values for either r, x , or y (whichever are given in the original problem) into the Pythagorean theorem. and then solve for the other variable, and then set up the r/x , r/y etc for the remaining trig functions.
(One other thing, about θ always being in quadrant one, the way I perceived it, is that in a related problem such as sinθ = -3/5 and cosθ=4/5 , it doesn't tell you what quadrant to put θ in so how do you determine this? )

 

[tiny-tim]

hi land_of_ice!

The problem is: cosθ=-4/5 θ in quadrant 3
…
Also, after I drew the picture, I never use it in the equation, I just plug in values for either r, x , or y (whichever are given in the original problem) into the Pythagorean theorem. and then solve for the other variable, and then set up the r/x , r/y etc for the remaining trig functions.

if you never used the picture, then of course you got it wrong!

there's always two answers for any cos or sin equation …

if cosθ is positive, then θ can be in quadrants 1 or 4; if cosθ is negative, then θ can be in quadrants 2 or 3 …

there is no way of knowing unless you are told in the question!

 

[land_of_ice]

hi land_of_ice!


if you never used the picture, then of course you got it wrong!

there's always two answers for any cos or sin equation …

if cosθ is positive, then θ can be in quadrants 1 or 4; if cosθ is negative, then θ can be in quadrants 2 or 3 …

there is no way of knowing unless you are told in the question!

Hey Tim, thanks for that , it was very helpful :)

 

[HallsofIvy]

Homework Statement

The problem is: cosθ=-4/5 θ in quadrant 3
and they ask that you find the exact value of the remaining trig functions,

Since you are told that this is in the third quadrant, One way to do this, since cosine is "near side over hypotenuse", is to draw a line from the origin of a coordinate system to (-4, 0), then a line down to (-4, -3) (you know the "opposite side" has length 3 by the Pythagorean theorem). Now sine is "opposite side over hyptenuse" or -3/5 (the hypotenuse in such a picture is always positive), tangent is "opposite side over near side" or -3/-4= 3/4, etc.

Homework Equations

I do not know all of the relevant things to do, which is why I have a question, about what they are basically, as far as quadrant is concerned, the sign of the answer and whether or not it matters that a picture is drawn and used or not.

The Attempt at a Solution

I got the exact opposite signs to my answer, as the book got, but the values were correct, also I don't know why but after doing about 15 of these problems, when I drew the picture (first step), the θ seemed to be in the same area of the unit circle every time , in quadrant 1, so how can you tell where θ is supposed to be? Also, after I drew the picture, I never use it in the equation, I just plug in values for either r, x , or y (whichever are given in the original problem) into the Pythagorean theorem. and then solve for the other variable, and then set up the r/x , r/y etc for the remaining trig functions.
(One other thing, about θ always being in quadrant one, the way I perceived it, is that in a related problem such as sinθ = -3/5 and cosθ=4/5 , it doesn't tell you what quadrant to put θ in so how do you determine this? )

If you are told that sin θ= -3/5 and cosθ= 4/5, then the "near side" is positive 4 so draw a line from the origin to (4, 0). The "opposite side" is negative 3 so draw a line from (4, 0) to (4, -3). That will be in the fourth quadrant. tangent and cotangent will be negative (specifically, -3/4 and -4/3), secant positive (5/4), and cosecant negative (-5/3).