# Vector Valued Function Using (or misusing) Trig Identity

1. Feb 20, 2012

### kmacinto

1. The problem statement, all variables and given/known data
The context of the problem is that it's a vector valued function (VVF) problem where I'm supposed to sketch a curve generated by a VVF. To make the sketching easier I'm supposed to convert a VVF to a real valued function so that I can take advantage of the shape of a curve covered in previous sections of the text in real valued function format.

2. Relevant equations
Sketch the curve represented by the vector valued function r(θ) and give the orientation of the curve. r(θ)=3sec(θ)i+2tan(θ)j.

3. The attempt at a solution
The real valued function that r(θ) converts to is a hyperbola given by the equation (x^2)/2=(y^2)/2+1

The steps given in the book are:
1) from r(θ) set:
x=3sec(θ), y=2tan(θ)

2) (x^2)/2=(y^2)/2+1 which is the solution.

My steps attempted are:
from: x=3sec(θ), y=2tan(θ)
1) divide x and y by the constants leaving:
x/3 = sec(θ) and y/2 = tan(θ)

2) equate both equations leaving:
x/3 + y/2 = sec(θ) + tan(θ)

3) square both sides leaving:
(x/3)^2 + (y/3)^2 = (sec(θ))^2 + (tan(θ))^2

and here's where I'm stuck...

Because both sides need a negative to work. For instance, for the (y/3)^2 to be on the right and positive it needs to be negative on the left and to get the right side to equal 1 using (tanθ)^2 + 1 = (secθ)^2, I need either the (tanθ)^2 or the (secθ)^2 to be negative where they're at now. Right?

Totally flustered!

Ken
PS. The attachment is the problem and given solution from the text book.

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Last edited: Feb 20, 2012
2. Feb 20, 2012

### tiny-tim

Hi Ken!
I don't understand what you're doing.

What is sec2θ - tan2θ ?

3. Feb 20, 2012

### HallsofIvy

Staff Emeritus
$(a+ b)^2\ne a^2+ b^2$