Why do we cancel sin by sin?

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 You're absolutely right! In trigonometry, sin(theta) represents a single function, not a product of "sin" and "theta". Here's why canceling sin by sin in physics equations works:

Trigonometric Identities:

In trigonometry, there are established relationships between trigonometric functions (sin, cos, tan, etc.) These relationships are called trigonometric identities and are always true regardless of the specific value of the angle (theta).

Example:

One such identity is sin^2(theta) + cos^2(theta) = 1 (where ^2 represents squaring). This identity essentially states that the square of sine (sin^2) added to the square of cosine (cos^2) will always equal 1, for any angle theta.

Canceling Sin in Physics Equations:

When you see sin appearing on both sides of a physics equation, you might be dealing with a situation where a trigonometric identity is being applied. By rearranging the equation and using a trigonometric identity, the sin terms might end up on opposite sides, allowing you to cancel them.

Example:

Imagine a physics equation involving a force (F) acting on an object at an angle (theta). The equation might involve a term like "F * sin(theta)". If another part of the equation also has a sin(theta) term, and you can manipulate the equation using a trigonometric identity, you might be able to isolate the desired variable and end up canceling the sin terms.

Key Points:

  • sin(theta) is a single unit, not a product.
  • Canceling sin terms works because of established trigonometric identities.
  • You're essentially using the properties of these functions to simplify the equation.

If you have a specific physics equation where you're unsure about canceling sin terms, feel free to share it! I can help you understand the underlying trigonometric identity.

sin (2A) = sin (A + 3)

 You're absolutely right! The sin terms in the equation sin(2A) = sin(A + 3) are not canceled out. Canceling mathematical functions in this way wouldn't be valid.

Here's what's actually happening:

This equation represents a specific trigonometric identity, not a cancellation.

Trigonometric identities are equations that relate trigonometric functions (sin, cos, tan, etc.) and are always true regardless of the specific value of the angle (A in this case).

The equation sin(2A) = sin(A + 3) is a specific example of the double-angle identity for sine, which can be written as:

sin(2A) = 2sin(A)cos(A)

This identity expresses the relationship between the sine of a doubled angle (2A) and the product of the sine and cosine of the original angle (A).

What's not happening:

  • The sin function is not being multiplied or divided by anything on the right side.
  • The sin(2A) term is not directly equal to the sin(A + 3) term.

What is happening:

  • The equation expresses a relationship between trigonometric functions.
  • Solving this equation for A would involve manipulating the equation using trigonometric properties, not canceling sin terms.

Why it might seem like canceling:

Since the equation has sin on both sides, it might appear tempting to cancel them. However, this is not a valid mathematical step.

Understanding the solution:

Solving this equation for A would involve using algebraic manipulations and other trigonometric identities. It's not a simple cancellation but a process of transforming the equation to isolate A.

Key takeaway:

  • sin(theta) is a single unit, not a product.
  • Trigonometric identities relate functions, not cancel them.

If you'd like to explore how to solve this specific equation for A, I can help you with the steps involved!

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