Find the equations to the straight lines passing through the following pairs of points.
`(a cos phi_(1), a sin phi_(1)) and (a cos phi_(2), a sin phi_(2))`

Answer

To find the equation of the straight line passing through the points \((a \cos \phi_1, a \sin \phi_1)\) and \((a \cos \phi_2, a \sin \phi_2)\), we can follow these steps:

### Step 1: Identify the coordinates
Let:
- Point A: \((x_1, y_1) = (a \cos \phi_1, a \sin \phi_1)\)
- Point B: \((x_2, y_2) = (a \cos \phi_2, a \sin \phi_2)\)

### Step 2: Use the point-slope form of the line
The equation of the line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) can be given by the formula:
\[ y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1) \]

### Step 3: Substitute the coordinates into the formula
Substituting our points into the formula:
\[ y - a \sin \phi_1 = \frac{(a \sin \phi_2 - a \sin \phi_1)}{(a \cos \phi_2 - a \cos \phi_1)} (x - a \cos \phi_1) \]

### Step 4: Simplify the equation
We can simplify the equation by factoring out \(a\) from the numerator and the denominator:
\[ y - a \sin \phi_1 = \frac{(a (\sin \phi_2 - \sin \phi_1))}{(a (\cos \phi_2 - \cos \phi_1))} (x - a \cos \phi_1) \]
This simplifies to:
\[ y - a \sin \phi_1 = \frac{\sin \phi_2 - \sin \phi_1}{\cos \phi_2 - \cos \phi_1} (x - a \cos \phi_1) \]

### Step 5: Use trigonometric identities
Using the trigonometric identities:
\[ \sin \phi_2 - \sin \phi_1 = 2 \cos\left(\frac{\phi_1 + \phi_2}{2}\right) \sin\left(\frac{\phi_2 - \phi_1}{2}\right) \]
\[ \cos \phi_2 - \cos \phi_1 = -2 \sin\left(\frac{\phi_1 + \phi_2}{2}\right) \sin\left(\frac{\phi_2 - \phi_1}{2}\right) \]
Substituting these identities into our equation gives:
\[ y - a \sin \phi_1 = \frac{2 \cos\left(\frac{\phi_1 + \phi_2}{2}\right) \sin\left(\frac{\phi_2 - \phi_1}{2}\right)}{-2 \sin\left(\frac{\phi_1 + \phi_2}{2}\right) \sin\left(\frac{\phi_2 - \phi_1}{2}\right)} (x - a \cos \phi_1) \]

### Step 6: Cancel terms
The \(2\) and \(\sin\left(\frac{\phi_2 - \phi_1}{2}\right)\) cancel out:
\[ y - a \sin \phi_1 = -\cot\left(\frac{\phi_1 + \phi_2}{2}\right) (x - a \cos \phi_1) \]

### Step 7: Rearranging the equation
Rearranging gives:
\[ \sin\left(\frac{\phi_1 + \phi_2}{2}\right)(y - a \sin \phi_1) + \cos\left(\frac{\phi_1 + \phi_2}{2}\right)(x - a \cos \phi_1) = 0 \]

### Final Equation
Thus, the equation of the straight line passing through the points \((a \cos \phi_1, a \sin \phi_1)\) and \((a \cos \phi_2, a \sin \phi_2)\) is:
\[ \sin\left(\frac{\phi_1 + \phi_2}{2}\right) y + \cos\left(\frac{\phi_1 + \phi_2}{2}\right) x - a \left(\sin\left(\frac{\phi_1 + \phi_2}{2}\right) \sin \phi_1 + \cos\left(\frac{\phi_1 + \phi_2}{2}\right) \cos \phi_1\right) = 0 \]