y
0 + y = y
2

cos(x) − sin(x)

give the general solution

To solve the trigonometric equation \( \cos(x) - \sin(x) = y \), we'll approach it systematically.

### Step 1: Express the equation in a more manageable form

First, recognize that the equation can be written as:
\[
\cos(x) - \sin(x) = y
\]

### Step 2: Transform the trigonometric expression

Let's transform the left-hand side using the identity for the sum and difference of sines and cosines. We can rewrite the expression as:
\[
\cos(x) - \sin(x) = \sqrt{2} \left( \frac{1}{\sqrt{2}}\cos(x) - \frac{1}{\sqrt{2}}\sin(x) \right)
\]

This simplifies to:
\[
\cos(x) - \sin(x) = \sqrt{2} \cos\left(x + \frac{\pi}{4}\right)
\]

Thus, the equation becomes:
\[
\sqrt{2} \cos\left(x + \frac{\pi}{4}\right) = y
\]

### Step 3: Solve for \( x \)

To solve for \( x \), divide both sides by \( \sqrt{2} \):
\[
\cos\left(x + \frac{\pi}{4}\right) = \frac{y}{\sqrt{2}}
\]

### Step 4: General solution

The general solution for \( \cos(\theta) = a \) is:
\[
\theta = 2n\pi \pm \arccos(a)
\]
where \( n \) is any integer.

Thus, the solution for \( x \) is:
\[
x + \frac{\pi}{4} = 2n\pi \pm \arccos\left(\frac{y}{\sqrt{2}}\right)
\]

Finally, subtract \( \frac{\pi}{4} \) from both sides to solve for \( x \):
\[
x = 2n\pi - \frac{\pi}{4} \pm \arccos\left(\frac{y}{\sqrt{2}}\right)
\]
where \( n \) is any integer.

### Summary of the General Solution:
\[
x = 2n\pi - \frac{\pi}{4} \pm \arccos\left(\frac{y}{\sqrt{2}}\right)
\]

### Questions and Further Exploration:
1. What is the specific solution when \( y = 0 \)?
2. How does the general solution change if \( y = \sqrt{2} \)?
3. Can this approach be applied to \( \cos(x) + \sin(x) = y \)?
4. How would you verify if a specific \( y \) value is valid for this equation?
5. How would the solution change if \( \cos(x) - \sin(x) = y \) is squared?

**Tip:** Remember, when solving trigonometric equations, transforming the expression into a more familiar form can simplify finding the general solution.

Learn how to solve the trigonometric equation \( \cos(x) - \sin(x) = y \) step-by-step with a detailed explanation. This problem involves applying trigonometric identities and finding the general solution for \( x \). Suitable for advanced high school students or those studying trigonometry.

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