How can we compute dx in terms of R and θ in this given right triangle?

[quantum13]

Homework Statement

This is part of a find the electric field problem. I've narrowed down the part I find confusing.

Consider a right triangle, with legs of length x and R. Angle θ is opposite x. Leg x is a segment of a ray starting where lines x and R intersect. There is a differential length dx along line x (it is at a vertex of the right triangle). Find dx in terms of R and θ.

Homework Equations

standard trig functions, pythagorean theorem?

The Attempt at a Solution

The answer according to the solutions manual is dx = (R/cos^2 θ) dθ. Obviously, I cannot understand at all where that came from.

I did get R tan ( θ + dθ) = x + dx but that isn't very helpful either

 

[Redbelly98]

It sounds like R is fixed, while x and θ can vary.

Can you write an expression for x in terms of R and θ?

 

[tomcue]

I would approach this problem by first understanding the given information and the goal of the problem. In this case, we are trying to find the electric field and we are given a right triangle with known dimensions and an angle θ.

To compute dx, we can use the trigonometric function tangent (tan). From the given information, we know that tan θ = x/R. Rearranging this equation, we get x = R tan θ.

Now, we can take the derivative of both sides to find dx in terms of R and θ. Using the chain rule, we get dx = R sec^2 θ dθ. This is almost the same as the solution provided in the solutions manual, except for the use of secant (sec) instead of cosine (cos). This is because secant is the inverse of cosine and is commonly used in finding derivatives of trigonometric functions.

Therefore, the final answer for dx is dx = (R sec^2 θ) dθ. This means that as θ increases, dx also increases at a rate proportional to R and sec^2 θ.

In summary, to compute dx in this problem, we used the given information and the trigonometric function tangent to find x in terms of R and θ. Then, we took the derivative to find the rate of change of x (dx) with respect to θ. This is a common approach in solving problems involving rates of change and trigonometry.