Solve cosxcos(-x) - sinxsin(-x) = 1

  • Thread starter ravey_dotson
  • Start date
In summary, the conversation is discussing how to simplify the expression cosxcos(-x) - sinxsin(-x) and how it relates to the angle addition formula. One person initially struggles with the concept, but eventually realizes that the expression can be simplified to cos(x)^2 + sin(x)^2 = 1 by using the identities cos(-x) = cos(x) and sin(-x) = -sin(x). Another person suggests an alternative method of using the angle addition formula, but notes that the signs must be corrected for it to work.
  • #1
ravey_dotson
3
0
cosxcos(-x) - sinxsin(-x) = 1

I know I'm making it harder than it is, but I can't seem to figure it out.
Any help would be appreciated.

RaveN
 
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  • #2
cos(-x)=cos(x). sin(-x)=-sin(x).
 
  • #3
When I plugged those in then I get cosx2 - sinx2 = 1 and then I get completely lost.
 
  • #4
When I plug them in I get cos(x)^2+sin(x)^2=1. There are two minus signs associated with the sin(x)^2.
 
  • #5
Because it's -sin to begin with. I knew I was making it too hard. Thanks.
 
  • #6
Another way: isn't the l.h.s. in the form of an angle addition formula?
 
  • #7
Not with the correct signs! It's cos2(x)+ sin2(x)!
 
  • #8
It is cos(x+(-x)). That is another way to do it.
 

1. What is the equation cosxcos(-x) - sinxsin(-x) = 1 asking to solve?

The equation is asking you to find the value(s) of x that make the equation true.

2. Why are there negative signs in the equation?

The negative signs represent the opposite trigonometric functions (cosine and sine) of x, which are reflected across the y-axis. This is due to the identity cos(-x) = cosx and sin(-x) = -sinx.

3. How do I solve this equation?

To solve this equation, you can use trigonometric identities, such as the Pythagorean identity (cos^2x + sin^2x = 1) and the double angle identity (cos2x = cos^2x - sin^2x), to simplify the equation and find the value(s) of x that make it true. You can also graph the equation and find the points of intersection with the line y = 1 to find the solutions.

4. Are there any restrictions on the values of x in this equation?

Yes, there are restrictions on the values of x in this equation. Since cosine and sine have a period of 2π, the equation will have infinitely many solutions. However, the values of x that make the equation true will be within the range of 0 to 2π, or 0 to 360 degrees.

5. Can I use a calculator to solve this equation?

Yes, you can use a calculator to solve this equation. However, it is important to make sure your calculator is set to the correct mode (degrees or radians) and to check for any restrictions on the values of x in the solution.

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