What Are the Solutions for cos(2x) + cos(x) = 0?

Click For Summary
SUMMARY

The equation cos(2x) + cos(x) = 0 has solutions within the interval 0 <= x <= 360 degrees. The valid solutions identified are x = 60 degrees, x = 180 degrees, and x = 300 degrees. The confusion regarding notation between cos(2x) and cos2x was clarified, emphasizing the importance of using parentheses for clarity. The quadratic form of the equation, rewritten as 2cos^2(x) - 1 = -cos(x), allows for easier factorization and solution identification.

PREREQUISITES
  • Understanding of trigonometric identities, specifically the double angle formula for cosine.
  • Familiarity with solving quadratic equations in trigonometric contexts.
  • Knowledge of the unit circle and angle measures in degrees.
  • Ability to manipulate and interpret mathematical expressions in text format.
NEXT STEPS
  • Study the double angle formula for cosine: cos(2x) = 2cos^2(x) - 1.
  • Learn how to solve trigonometric equations using factorization techniques.
  • Explore the unit circle to understand angle measures and their corresponding cosine values.
  • Practice writing mathematical expressions clearly to avoid notation confusion.
USEFUL FOR

Students studying trigonometry, educators teaching mathematical notation, and anyone looking to improve their problem-solving skills in trigonometric equations.

Helly123
Messages
581
Reaction score
20

Homework Statement


cos2x + cos x = 0 (0 <= x <= 360)

Homework Equations

The Attempt at a Solution


cos2x + cos x = 0
2cos(3x)/2 cos(x)/2 = 0

3x/2 = 90 degrees
x = 60 degrees

x/2 = 90
x = 180

3x/2 = 270
x = 180

x/2 = 270
x = 540 (not qualified)

is there any more possibility (answers) for x?
 
Physics news on Phys.org
Helly123 said:
cos2x + cos x = 0 (0 <= x <= 360)
Is it cos(2x) or cos2x?
 
cnh1995 said:
Is it cos(2x) or cos2x?
it's cos2x
 
Helly123 said:
it's cos2x
Of course:-p!
I hadn't read your complete solution.

But I think you've found all the possible values for x.
 
cnh1995 said:
Of course:-p!
I hadn't read your complete solution.

But I think you've found all the possible values for x.
thank you. But there's one more answers but I don't know how to find it
 
Check the 270...
 
BvU said:
Check the 270...
I see, I get it. thanks
 
Helly123 said:

Homework Statement


cos2x + cos x = 0 (0 <= x <= 360)

Homework Equations

The Attempt at a Solution


cos2x + cos x = 0
2cos(3x)/2 cos(x)/2 = 0

3x/2 = 90 degrees
x = 60 degrees

x/2 = 90
x = 180

3x/2 = 270
x = 180

x/2 = 270
x = 540 (not qualified)

is there any more possibility (answers) for x?
What about 300°?
 
Why didn't you just write $$\cos{2x}=2\cos^2{x}-1$$and then directly solve the resulting (factorable) quadratic equation for ##\cos{x}##
 
  • #10
ehild said:
What about 300°?
Yes, that's the answer
 
  • #11
Chestermiller said:
Why didn't you just write $$\cos{2x}=2\cos^2{x}-1$$and then directly solve the resulting (factorable) quadratic equation for ##\cos{x}##
I see.. I didn't realize it
 
  • #12
Helly123 said:
cos2x + cos x = 0 (0 <= x <= 360)

cnh1995 said:
Is it cos(2x) or cos2x?
To prevent such confusion, use parentheses -- i.e., write cos(2x) instead of cos2x. Also, we get a lot of students who are new to writing math expressions as text. I've seen many who write cos2x when they really mean cos2(x). For exponents, the ^ symbol is often used.
 
  • #13
Mark44 said:
To prevent such confusion, use parentheses -- i.e., write cos(2x) instead of cos2x. Also, we get a lot of students who are new to writing math expressions as text. I've seen many who write cos2x when they really mean cos2(x). For exponents, the ^ symbol is often used.
Ok I'll be more careful
 

Similar threads

Trigonometric equation solving 2cos x=tan x
  • · Replies 6 ·
Replies
6
Views
3K
Solving a trigonometric equation with multiples of ##\tan##
  • · Replies 7 ·
Replies
7
Views
2K
Trigonometric Equations Problems - Rather Confused
  • · Replies 1 ·
Replies
1
Views
2K
Solving a Trigonometric Equation
  • · Replies 3 ·
Replies
3
Views
2K
Solving a nested logarithmic equation
  • · Replies 2 ·
Replies
2
Views
2K
Trigonometric Equation Solving: Cosine and Sine Identities for Homework
  • · Replies 5 ·
Replies
5
Views
2K
Using Power Reducing Formulas to rewrite Trig Expressions
  • · Replies 7 ·
Replies
7
Views
2K
Solving a trigonometric equation for the angle
  • · Replies 3 ·
Replies
3
Views
2K
Solve for all angles x: cos(2x) + cos(x) = 0, where 0<x<2pi
  • · Replies 25 ·
Replies
25
Views
3K
How to find a constant in this quadratic equation?
  • · Replies 10 ·
Replies
10
Views
4K