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A trig equation

  • #1

Homework Statement


Find all solutions of the equation (tanx + sqrt(3))(cosx+2)=0


Homework Equations


Any of the identities


The Attempt at a Solution



(tanx + sqrt(3))(cosx + 2)=0
tan(x)cos(x)+2tan(x) + sqrt(3)cos(x) + 2sqrt(3) = 0
(sin(x)/cos(x))cos(x) + 2tan(x) + sqrt(3)cos(x) + 2sqrt(3) = 0
sin(x) + 2tan(x) + sqrt(3)cos(x) + 2 sqrt(3) = 0
sin^2(x) + 4tan^2(x) + 3cos^2(x) + 12 = 0 (square)
sin^2(x) + cos^2(x) + 2cos^2(x) + 4tan^2(x) + 12 = 0
2cos^2(x) + 4tan^2(x) + 13 = 0
cos^2(x) + 2tan^2(x) + 6.5 = 0 (Divide by 2)
1 - sin^2(x) + 2tan^2(x) + 6.5 = 0 (sin^2(x) + cos^2(x) = 1; cos^2(x) = 1 - sin^2(x))
-sin^2(x) + 2tan^2(x) + 7.5 = 0
-1 + 2sin^2(x) + 7.5/sin^2(x) (Multiply by sin^2(x))
2sin^2(x) + (6.5 + sin^2(x) + cos^2(x))/sin^2(x) -1 = 0
2sin^2(x) + 6.5 - 1 + cos^2(x) = 0
sin^2(x) + sin^2 (x) + cos^2(x) + 5.5 = 0
sin^2(x) + 6.5 = 0
sin(x) + sqrt(6.5) = 0
sin(x) = -sqrt(6.5)

The book gives an answer of -(pi/3) + k(pi), which I'd think is correct over my equations - I honestly don't know how to get to an answer because the book is horrid at teaching and I'm out of school for a bit so I can't ask the teacher for help.
 

Answers and Replies

  • #2
I didn't really look at your work too closely, but this way seems more obvious to me.

(tan(x) + sqrt(3))(cos(x) + 2) = 0

This is equation is factored, i.e. in the form you need to solve the problem.

Therefore, this equation equals 0 when:
1) tan(x) + sqrt(3) = 0
or
2) cos(x) + 2 = 0

1) check values of 'x' for which
tan(x) = -sqrt(3)
is true
*can you solve for x?*

2) check values of 'x' for which
cos(x) = -2
is true
*can you sovle for x?*
 
  • #3
1,631
4
ey body, u can do this pretty simply,

(tanx + sqrt(3))(cosx+2)=0

when this is zero?

it is zero when (tanx + sqrt(3))=0 or (cosx+2)=0, or both of them are zero, so what u have to do is just solve these two equations now, and find the overall answer.

sorry, chaotic allready stated this. i did not see it
 
  • #4
HallsofIvy
Science Advisor
Homework Helper
41,808
933
I guarentee that if you put "Solve (x- 3)(x- 5)= 0" on a basic algebra test, 1/3 of the students will multiply it out and use the quadratic formula to solve x2- 8x+ 15= 0!
 
  • #5
symbolipoint
Homework Helper
Education Advisor
Gold Member
5,853
1,036
HallsofIvy,

Sometimes people forget the simple stuff and use the less simple stuff; they may forget to recognize what they have already studied. Usually, the students still UNDERSTAND the simple stuff, but they just forgot that this simple stuff is all that is needed.
 
  • #6
That's what happened with me. Too used to doing complex equations that I forgot that there were stupidly simple ways of doing it.

Thanks for all the help!
 

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