Find the values of x and y, lying between 0 and 360 which satisfy the equations,
`cos((1)/(2)y^(@)+71^(@))=-0.3420`.

Answer

To solve the equation \( \cos\left(\frac{1}{2}y + 71^\circ\right) = -0.3420 \), we will follow these steps:

### Step 1: Set up the equation
We start with the equation:
\[ \cos\left(\frac{1}{2}y + 71^\circ\right) = -0.3420 \]

### Step 2: Find the reference angle
To find the angle whose cosine is \(-0.3420\), we can use the inverse cosine function:
\[ \theta = \cos^{-1}(-0.3420) \]
Calculating this gives:
\[ \theta \approx 109.99^\circ \quad \text{(1st quadrant reference)} \]
Since cosine is negative, we also consider the second quadrant:
\[ \theta' = 360^\circ - 109.99^\circ = 250.01^\circ \]

### Step 3: Set up the equations
Now, we have two cases for the angle:
1. \(\frac{1}{2}y + 71^\circ = 109.99^\circ\)
2. \(\frac{1}{2}y + 71^\circ = 250.01^\circ\)

### Step 4: Solve for \(y\) in each case

**Case 1:**
\[ \frac{1}{2}y + 71^\circ = 109.99^\circ \]
Subtract \(71^\circ\) from both sides:
\[ \frac{1}{2}y = 109.99^\circ - 71^\circ \]
\[ \frac{1}{2}y = 38.99^\circ \]
Multiply both sides by 2:
\[ y = 77.98^\circ \]

**Case 2:**
\[ \frac{1}{2}y + 71^\circ = 250.01^\circ \]
Subtract \(71^\circ\) from both sides:
\[ \frac{1}{2}y = 250.01^\circ - 71^\circ \]
\[ \frac{1}{2}y = 179.01^\circ \]
Multiply both sides by 2:
\[ y = 358.02^\circ \]

### Step 5: Final values of \(y\)
The values of \(y\) that satisfy the equation are:
\[ y \approx 77.98^\circ \quad \text{and} \quad y \approx 358.02^\circ \]

### Step 6: Determine \(x\)
Since the problem does not specify a relationship for \(x\), we can assume \(x\) can take any value between \(0^\circ\) and \(360^\circ\).

### Summary of Solutions
The values of \(y\) are:
- \(y \approx 77.98^\circ\)
- \(y \approx 358.02^\circ\)