Trying to understand why a trigonometric substitution for the x value in an equation describing a circle uses Sin($\theta$).

Question

Essentially I am trying to solve a definite integral for an equation that describes a circle.

I have $\sqrt(R^2 - (x - A)^2) - B)$ (A and B being the circle origin offsets and R the radius)

At a point in the solution I substitute for (x -A) the function Rcos($\theta$), but everywhere on the internet uses Rsin($\theta$), i understand why sin is prefered I do not understand why sin is allowed.

As far as I can tell Rcos($\theta$) seems correct for a circle as it describes the x value of the right triangle that is formed by R,x,y with angle Rx being $\theta$. I do not follow why sin and cos are interchangeable and I am thoroughly confused.

Any help is appreciated. Thanks.

Answer 1

It seems like you're trying to understand why the trigonometric functions sine and cosine are interchangeable in certain contexts. In trigonometry, sine and cosine are related through the unit circle. Since the unit circle has radius 1, the x-coordinate of a point on the unit circle corresponds to cosine and the y-coordinate corresponds to sine.

When you're dealing with a circle with radius R centered at the origin, the substitution x = Rcos(θ) and y = Rsin(θ) comes from considering the parametric equations of the circle. Both sine and cosine are used because they describe the relationship between the angle θ and the x and y coordinates of a point on the circle.

So, it's not that one is "allowed" while the other isn't, but rather that both are valid representations of the relationship between the angle θ and the coordinates of a point on the circle. Depending on the context or preference, one might be more convenient to use over the other. I hope this explanation help

Answer 2

I understand your confusion. Let me clarify.

When substituting for x in the equation of a circle using polar coordinates, we use R cos θ because the x-coordinate of a point on the circle can be represented by R cos θ, where R is the radius and θ is the angle.

However, when solving integrals or equations involving circles and polar coordinates, it’s common to use the identity x = R sin θ as well. This might seem counterintuitive at first since R sin θ represents the y-coordinate of a point on the circle, not the x-coordinate.

The reason for using R sin θ as a substitution for x lies in the symmetry of the circle. The equation of a circle is symmetric with respect to both the x-axis and the y-axis. Therefore, if x = R cos θ represents one half of the circle (e.g., the right half), then x = -R cos θ represents the other half (e.g., the left half). Similarly, y = R sin θ represents one half of the circle (e.g., the top half), and y = -R sin θ represents the other half (e.g., the bottom half).

So, while R sin θ technically represents the y-coordinate, we can still use it as a substitution for x by taking into account the symmetry of the circle. This allows us to simplify calculations and expressions involving circles in polar coordinates. I hope this time your confusion is cleared

Answer 3

The reason Rsin($\theta$) can be used is because we choose to measure the angle from the y axis instead of the conventional way from the x axis.

If you do this your x value becomes RSin($\theta$) instead of RCos($theta$).

Thanks for responders help