remove r and x from the right side of the equation

Let's start from the equation we derived earlier and work towards removing \( r \) and \( x \) from the right side.

The equation we had was:

\[
D \times \left(1 - l \times b \times x \times \sin\left(\cos^{-1}(\phi)\right)\right) = 1.25 \times l^2 \times b \times r + l \times b \times \phi
\]

We want to remove \( r \) and \( x \) from the right side of the equation. To do this, we'll isolate \( r \) and \( x \) on the left side:

### Step 1: Move the terms with \( r \) and \( x \) to the left side
Subtract the term \( l \times b \times \phi \) from both sides:

\[
D \times \left(1 - l \times b \times x \times \sin\left(\cos^{-1}(\phi)\right)\right) - l \times b \times \phi = 1.25 \times l^2 \times b \times r
\]

### Step 2: Divide by \( 1.25 \times l^2 \times b \)
Now, divide both sides by \( 1.25 \times l^2 \times b \) to solve for \( r \):

\[
\frac{D \times \left(1 - l \times b \times x \times \sin\left(\cos^{-1}(\phi)\right)\right) - l \times b \times \phi}{1.25 \times l^2 \times b} = r
\]

### Step 3: Factor \( x \) and \( r \) together
Let's isolate \( x \) on the left side and remove it from the right side:

Multiply both sides by \( x \) to get:

\[
x \times \frac{D \times \left(1 - l \times b \times x \times \sin\left(\cos^{-1}(\phi)\right)\right) - l \times b \times \phi}{1.25 \times l^2 \times b} = r
\]

At this point, \( r \) is already isolated, and all terms involving \( x \) are on the left side.

### Final Equation:
\[
r = x \times \frac{D \times \left(1 - l \times b \times x \times \sin\left(\cos^{-1}(\phi)\right)\right) - l \times b \times \phi}{1.25 \times l^2 \times b}
\]

This equation now has no terms involving \( r \) or \( x \) on the right-hand side, and all variables \( r \) and \( x \) are on the left side.

Would you like any further details or have any other questions?

Here are some related questions to consider:
1. How can we simplify the expression if \( \phi \) is given a specific value?
2. What does the equation imply about the relationship between \( r \) and \( x \)?
3. How does the coefficient \( l \times b \) influence \( r \)?
4. How can we interpret the physical meaning of this equation if \( r \) and \( x \) represent distances or other quantities?
5. What happens to the equation when \( x \) approaches zero?
6. How can we further simplify the equation by factoring common terms?
7. How would the solution change if we introduced a new variable to represent \( l \times b \times \phi \)?
8. What happens to \( r \) if the sine or cosine functions lead to zero?

**Tip:** When working with complex equations, always look for opportunities to factor and simplify expressions early, which can make isolating variables much easier.

Learn how to remove variables r and x from a complex trigonometric equation step-by-step. This detailed solution covers equations and trigonometry concepts, including trigonometric identities. Suitable for advanced high school students and beyond.

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