Thread: Finding Trig functions using the Triangle Method (due tommorow)

Problem:


That -1 is suppose to represent the inverse. Basically, I've narrowed things down the 4th quadrant (ASTC), and I've reduced the problem down to:



Alright, we also have Opposite over Adjacent as well (TOA), so therefore what would we be working with to create the formula (Pythagorean Theorem) needed to solve the question? Or to make things more precise, how exactly would this triangle and equation look if everything were to be plugged in correctly given the details I have gathered about the problem? Furthermore, what roll is the secant function going to play when I finally solve the inside of the inverse tangent function?

And finally, what is the difference between what is being asked of the problems... one is asking for the exact evaluation, while the other is asking for an identity. Our professor hasn't gotten much into depth of any of this stuff, but I got a worksheet due tomorrow on it, so I'm kind of on my own here. And I also apologize if I'm not clear at all on my explanations, as I'm extremely tired as I type this. I probably should used some more mathematical terminology, but my mind is so worn-out right now it's a pretty rough thing to come by.

sec[arctan(-2)]

sec is a trig function.

arctan(-2) is an angle, believe it or not. Let as call it theta.
It is an angle whose tan function value is -2 ........a negative value.
Tangent is negative in the 2nd and 4th quadrant, so arctan(-2), or theta, is in the 2nd or 4th quadrant.

In the 2nd quadrant, the opp side is positive, while the adj side is negative.
so, tan(theta) = -2 = (2) / (-1).
Meaning, the opp is 2, and the adj is -1.

You are asked to find sec(theta).
Sec = hyp / adj
You don't know the hypotenuse. So you solve for it by using the Pythagorean theorem.
(hyp)^2 = (opp)^2 +(adj)^2
(hyp)^2 = (2)^2 +(-1)^2 = 5
hyp = sqrt(5) -------------hyp is always positive.
So,
sec(theta) = sec[arctan(-2)] = sqrt(5) / (-1) = -sqrt(5) ------answer, in the 2nd quadrant.

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EDIT:
Ooppsss, you want the theta in the 4th quadrant?

Then, opp is negative , while adj is positive
tan(theta) = -2 = (-2)/1

hyp = sqrt(5) .........always positive

So,
sec(theta) = sec[arctan(-2)] = hyp/adj = sqrt(5) / 1 = sqrt(5) .........answer, in the 4th quadrant.

Originally Posted by ticbol

sec[arctan(-2)]

sec is a trig function.

arctan(-2) is an angle, believe it or not. Let as call it theta.
It is an angle whose tan function value is -2 ........a negative value.
Tangent is negative in the 2nd and 4th quadrant, so arctan(-2), or theta, is in the 2nd or 4th quadrant.

In the 2nd quadrant, the opp side is positive, while the adj side is negative.
so, tan(theta) = -2 = (2) / (-1).
Meaning, the opp is 2, and the adj is -1.

You are asked to find sec(theta).
Sec = hyp / adj
You don't know the hypotenuse. So you solve for it by using the Pythagorean theorem.
(hyp)^2 = (opp)^2 +(adj)^2
(hyp)^2 = (2)^2 +(-1)^2 = 5
hyp = sqrt(5) -------------hyp is always positive.
So,
sec(theta) = sec[arctan(-2)] = sqrt(5) / (-1) = -sqrt(5) ------answer, in the 2nd quadrant.

That in particular cleared up a lot. Thanks for your help!