Hi! Why am I getting a rational answer in radians, but the book says degrees? *trigonometric equations*?

Question

Hi, please tell me what I'm doing wrong because I can't figure it out! I'm supposed to solve this trigonometric equation on [0, 2pi).

It is:

6cos²x + 5cosx + 1 = 0

I figured that I'd solve it with the quadratic, simple enough.

-5 +/- √(25 - (4)(6)(1)
------ --------- --------- --
12

=

-5 + 1
---------
12

and

-5-1
-------
12

Giving me -1/3 and -1/2. I then discarded the -1/3 (which I apparently shouldn't have), and got values of 7pi/6 and 11pi/6 for the -1/2.

What I'm thinking is that I should have used a calculator and done the arccos for both (-1/3) and (-1/2). Would that be the right thing to do?

Thanks so much for any help! I'll pick a best answer TODAY!!!!!

Answer 1

I'm really not sure what the problem is as you seem to have it all worked out. You just appear to be stumbling on the range where it is given in radians but you are supposed to answer in degrees?

What you have is cosx = -1/2 and cosx = -1/3

Each gives 2 results on [0,2pi] and you will get it using the calculator and a simple trigonometric identity which is (in radians)

cos(x) = cos(2pi - x) and in degrees it is cos(x) = cos(360-x).

Starting with

arccos(-1/2) = 2pi/3 or 120 degrees

This will also be true for 360 - 120 = 240 degrees.

Now do the same for arccos(-1/3) and you're set (it won't be a round number)..

Answer 2

Calculator or trig tables for arccos of -1/3 would be a good start. But you know that cos is negative in the 2nd and 3rd quadranta (All Students Take Calculus...All trig functions are positive in the first quadrant, sin and cosecant are positive in the 2nd, tan and cotan in the third and cos and sec in the 4th)
arc cos of 1/2 is 60 deg (1st quad); arc cos (-1/2) is120 in the 2nd quad and 240 deg in the third, and arc cos (1/2) if the 4th quad is 300 deg. Since you are looking at arc cos (-1/2) you may discard the 1st and 4th quad where cos is positive.
You know there are 2pi radians in a circle, and if you wish you may use a proportion to solve for number of radians in 120 deg and 240 deg. 2pi radians/360 = x radians/120
and 2pi radians/360 = x radians/240 so x = 4/3 pi radians.

I do think your initial equation is more easily solved by factoring 6cos^2x + 5cosx + 1 = 0
(3cosx +1)(2cosx + 1) = 0
setting each factor equal to zero, you have cosx = -1/2 and cosx = -1/3 and proceed to find the arc cos x for each value.

Answer 3

Here's how I would do it.

First, to avoid confusion, I'd substitute a new variable z for cos(x) then, the equation becomes...
6z^2 + 5z + 1 = 0

You get z=-1/3 and -1/2

Now, you know z=cos(x)
So -1/3 = cos(x) and -1/2=cos(x)

Then evaluating each individually by a calculator (be sure to set your calculator in DEGREE MODE!)
-1/3=cos(x)
x=arccos(-1/3)=109.5 degrees

But wait... be careful of the domain/range for the arccosine function! The value should appear in second and the third quadrant but the arccos only returns value in first and second. So you have to evaluate the same for the 3rd.

If you don't understand this, go back to the unit circle and see how this is possible.

x2=360-109.5
x2=250.5 degrees

Now, you have to do the same for the other answer, x=-1/2.
Don't forget the domain of arccos function, and be sure to put your calculator in the correct mode.

P.S. It is strange that your book gives you the range in radian but expects answer in degrees. When giving an answer, MAKE SURE you say degrees or put a degree symbol. Otherwise, it is considered radian since radian is a unitless measurement.

Answer 4

human beings do not continually agree and at the same time as they see something that they don't believe, they offer a thumbs down. I mean, look on the religion questions...if an atheists asks then each and every of the Christians get thumbs down, if a Christian asks, the Atheists get the thumbs down. i wager that is like somewhat skill vacation or something.

Answer 5

cos x = -1/3 indicates x = 70.5degrees if cos x is positive i.e 109.5 degrees since cos x is negative
cos x = -1/2 indicates x = 120 degrees since cos x is negative