Find the equation of the straight lines joining the points `(acostheta_(1),asintheta_(1))` and `(acostheta_(2),asintheta_(2))`.

Answer

To find the equation of the straight line joining the points \((a \cos \theta_1, a \sin \theta_1)\) and \((a \cos \theta_2, a \sin \theta_2)\), we can follow these steps:

### Step 1: Identify the Points
Let the points be:
- Point A: \( (x_1, y_1) = (a \cos \theta_1, a \sin \theta_1) \)
- Point B: \( (x_2, y_2) = (a \cos \theta_2, a \sin \theta_2) \)

### Step 2: Calculate the Slope of the Line
The slope \( m \) of the line joining two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the coordinates of points A and B:
\[ m = \frac{a \sin \theta_2 - a \sin \theta_1}{a \cos \theta_2 - a \cos \theta_1} \]
Factoring out \( a \):
\[ m = \frac{a (\sin \theta_2 - \sin \theta_1)}{a (\cos \theta_2 - \cos \theta_1)} = \frac{\sin \theta_2 - \sin \theta_1}{\cos \theta_2 - \cos \theta_1} \]

### Step 3: Use the Point-Slope Form of the Line
The equation of the line in point-slope form is:
\[ y - y_1 = m(x - x_1) \]
Substituting \( y_1 = a \sin \theta_1 \), \( m \), and \( x_1 = a \cos \theta_1 \):
\[ y - a \sin \theta_1 = \frac{\sin \theta_2 - \sin \theta_1}{\cos \theta_2 - \cos \theta_1} (x - a \cos \theta_1) \]

### Step 4: Rearranging the Equation
Cross-multiplying gives:
\[ (y - a \sin \theta_1)(\cos \theta_2 - \cos \theta_1) = (\sin \theta_2 - \sin \theta_1)(x - a \cos \theta_1) \]
Expanding both sides:
\[ y \cos \theta_2 - y \cos \theta_1 - a \sin \theta_1 \cos \theta_2 + a \sin \theta_1 \cos \theta_1 = x \sin \theta_2 - x \sin \theta_1 - a \cos \theta_1 \sin \theta_2 + a \cos \theta_1 \sin \theta_1 \]

### Step 5: Collecting Terms
Rearranging gives:
\[ y \cos \theta_2 - x \sin \theta_2 + a (\sin \theta_1 \cos \theta_1 - \sin \theta_1 \cos \theta_2 + \cos \theta_1 \sin \theta_2) = 0 \]

### Step 6: Final Equation
Using trigonometric identities, we can simplify further. The final equation of the line can be expressed as:
\[ x \cos \left(\frac{\theta_1 + \theta_2}{2}\right) + y \sin \left(\frac{\theta_1 + \theta_2}{2}\right) = a \left( \cos \left(\frac{\theta_1 - \theta_2}{2}\right) \right) \]