Angle Calculation: Sum and Difference

The problem asks for the value of the greater angle in radians, given the sum of two angles ($155^\circ$) and their difference ($\frac{\pi}{2}$). We need to solve a system of equations after ensuring consistent units.

Step 1: Unit Conversion

First, convert the sum of the angles from degrees to radians:

Sum in radians = $155^\circ \times \frac{\pi}{180^\circ} = \frac{155\pi}{180} = \frac{31\pi}{36}$ radians.

The difference is already given in radians: $\frac{\pi}{2}$.

Step 2: Setting Up Equations

Let the two angles be $x$ and $y$, where $x$ is the greater angle. We can set up two equations based on the given information:

• Equation 1 (Sum): $x + y = \frac{31\pi}{36}$
• Equation 2 (Difference): $x - y = \frac{\pi}{2}$

Step 3: Solving for the Greater Angle

To find the value of the greater angle ($x$), we can add the two equations together:

$(x + y) + (x - y) = \frac{31\pi}{36} + \frac{\pi}{2}$

Combine like terms:

$2x = \frac{31\pi}{36} + \frac{18\pi}{36}$

Add the fractions (using a common denominator of 36):

$2x = \frac{49\pi}{36}$

Isolate $x$ by dividing both sides by 2:

$x = \frac{49\pi}{36 \times 2}$

$x = \frac{49\pi}{72}$

Therefore, the value of the greater angle is $\frac{49\pi}{72}$ radians.