To solve for the angle \( a \) in the equation \( \cos(a) + x = \sin(a) + y \), follow these steps:

### Step 1: Rearrange the equation
Move all terms involving \( a \) to one side of the equation and the constants to the other:
\[
\cos(a) - \sin(a) = y - x
\]

### Step 2: Use a trigonometric identity
Recognize that the left side can be rewritten using the identity for the difference of cosines and sines:
\[
\cos(a) - \sin(a) = \sqrt{2} \cos\left(a + \frac{\pi}{4}\right)
\]
Thus, the equation becomes:
\[
\sqrt{2} \cos\left(a + \frac{\pi}{4}\right) = y - x
\]

### Step 3: Solve for \( a \)
Divide both sides by \( \sqrt{2} \):
\[
\cos\left(a + \frac{\pi}{4}\right) = \frac{y - x}{\sqrt{2}}
\]
Now, take the inverse cosine (arccos) to solve for \( a + \frac{\pi}{4} \):
\[
a + \frac{\pi}{4} = \arccos\left(\frac{y - x}{\sqrt{2}}\right)
\]
Finally, solve for \( a \):
\[
a = \arccos\left(\frac{y - x}{\sqrt{2}}\right) - \frac{\pi}{4}
\]

### Final Answer:
\[
a = \arccos\left(\frac{y - x}{\sqrt{2}}\right) - \frac{\pi}{4}
\]

Would you like more details or have any other questions?

Here are 5 related questions:
1. How do you apply the sum and difference formulas for sine and cosine?
2. What is the geometric interpretation of the trigonometric identity used?
3. How do you calculate the arccosine of a value on a calculator?
4. What are the possible ranges of values for \( x \) and \( y \) in this equation?
5. Can this equation have more than one solution for \( a \)?

**Tip:** When solving trigonometric equations, always consider the domain and range of the trigonometric functions involved to ensure that your solutions are valid.

Learn how to solve the trigonometric equation cos(a) + x = sin(a) + y step-by-step with detailed explanations and examples. This problem involves trigonometric identities and inverse trigonometric functions, suitable for advanced high school and college-level students.

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