Proving: 3A = (cos(2A)-cos(4A))/(sin(4A)-sin(2A))

  • Thread starter LoveandHate
  • Start date
Use the sum and difference identities for cosine and sine to rewrite the numerator and denominator in terms of 3A. Then, cancel out the common factors and simplify to get the left side, which is tan(3A). So, the identity is proven.
  • #1
LoveandHate
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Homework Statement


I have to prove that 3A=(cos(2A)-cos(4A))/(sin(4A)-sin(2A))


Homework Equations


All of the trig identites.


The Attempt at a Solution


Well, I know I can expand tan(3A) to sin(3A)/cos(3A). After this, I have no idea what to do. I do not know any way to simplify or expand the right side of the equation. If someone can help me out by just showing the me the first step, if possible, then I can try to solve the rest of the identity by myself.
Thanks!
 
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  • #2
LoveandHate said:

Homework Statement


I have to prove that 3A=(cos(2A)-cos(4A))/(sin(4A)-sin(2A))


Homework Equations


All of the trig identites.


The Attempt at a Solution


Well, I know I can expand tan(3A) to sin(3A)/cos(3A). After this, I have no idea what to do. I do not know any way to simplify or expand the right side of the equation. If someone can help me out by just showing the me the first step, if possible, then I can try to solve the rest of the identity by myself.
Thanks!
I don't think you have written the problem correctly. I believe you meant to write
tan(3A) = (cos(2A)-cos(4A))/(sin(4A)-sin(2A))

If so, start with the right side and notice that 2A = 3A - A, and 4A = 3A + A.
 

FAQ: Proving: 3A = (cos(2A)-cos(4A))/(sin(4A)-sin(2A))

How do you prove the equation 3A = (cos(2A)-cos(4A))/(sin(4A)-sin(2A))?

To prove this equation, we can use the trigonometric identities of double angle and half angle formulas. We can rewrite the right side of the equation as (2cos²(A) - 1)/(2sin(A)cos(A)), and then use the identity sin²(A) + cos²(A) = 1 to simplify it to just 1/2. Therefore, the equation is proven to be true.

What is the purpose of proving this equation?

Proving this equation allows us to understand the relationship between trigonometric functions and how they can be simplified. It also helps us to solve more complex equations and problems involving trigonometry.

Can this equation be used in real-world applications?

Yes, this equation can be used in various real-world applications such as engineering, physics, and navigation. It can be applied to solve problems involving angles, distances, and other geometric measurements.

Are there any limitations to this equation?

Like any mathematical equation, there are limitations to this equation. It may not be applicable in certain situations where the input values do not meet the necessary conditions, such as when the denominator (sin(4A)-sin(2A)) equals zero.

Are there any other ways to prove this equation?

Yes, there are other ways to prove this equation. Some may use the sum and difference identities or the Pythagorean identities. The method used may vary depending on the individual's preference or the specific problem being solved.

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