Given an equation sin theta=cos theta tan theta.
what does this mean dimensionally?
given:
sinθ = cosθtanθ
sinθ = cosθ(sinθ/cosθ)
sinθ = sinθ
yup, I really believe that.
What you are looking at in
sinθ = cosθtanθ
is what is called an "identity"
thanks.
To understand the dimensional meaning of the equation sin(theta) = cos(theta) * tan(theta), we need to consider the dimensions of each trigonometric function involved.
Let's assign a dimension symbol [ ] to represent the dimension of a quantity. For example, [θ] represents the dimension of angle.
The dimensions of the trigonometric functions are as follows:
- sin(theta): [sin(theta)] = dimensionless (no dimension)
- cos(theta): [cos(theta)] = dimensionless (no dimension)
- tan(theta): [tan(theta)] = dimensionless (no dimension)
Since sin(theta) and cos(theta) have the same dimension and are multiplied together, their product remains dimensionless. Therefore, we have:
[sin(theta)] = [cos(theta)] * [tan(theta)]
This equation means that the dimension of the left-hand side (LHS), which is dimensionless, must be equal to the dimension of the right-hand side (RHS), which is also dimensionless. This condition is always satisfied because both sides are dimensionless.
In conclusion, the equation sin(theta) = cos(theta) * tan(theta) means that both sides of the equation have no dimensions.