If sin(θ)/x = cos(θ)/y, determine the value of sin(θ) + cos(θ) in terms of x and y.

We are given the equation:

\[
\frac{\sin(\theta)}{x} = \frac{\cos(\theta)}{y}
\]

To solve for \(\sin(\theta) + \cos(\theta)\) in terms of \(x\) and \(y\), follow these steps:

### Step 1: Cross-multiply
Cross-multiplying the given equation:

\[
\sin(\theta) \cdot y = \cos(\theta) \cdot x
\]

This can be rewritten as:

\[
\sin(\theta) y = \cos(\theta) x
\]

### Step 2: Express \(\sin(\theta)\) in terms of \(\cos(\theta)\)
Rearrange the equation to express \(\sin(\theta)\) in terms of \(\cos(\theta)\):

\[
\sin(\theta) = \frac{x}{y} \cos(\theta)
\]

### Step 3: Express \(\sin(\theta) + \cos(\theta)\)
Now, substitute the expression for \(\sin(\theta)\) into \(\sin(\theta) + \cos(\theta)\):

\[
\sin(\theta) + \cos(\theta) = \frac{x}{y} \cos(\theta) + \cos(\theta)
\]

Factor out \(\cos(\theta)\):

\[
\sin(\theta) + \cos(\theta) = \cos(\theta) \left(\frac{x}{y} + 1\right)
\]

### Step 4: Express \(\cos(\theta)\) in terms of \(y\)
To find \(\cos(\theta)\) in terms of \(y\), use the original equation:

\[
\cos(\theta) = \frac{y}{x} \sin(\theta)
\]

Substituting into the equation:

\[
\sin(\theta) + \cos(\theta) = \frac{y}{x} \sin(\theta) \left(\frac{x}{y} + 1\right)
\]

Simplifying, we have:

\[
\sin(\theta) + \cos(\theta) = \sin(\theta) \left(1 + \frac{y}{x}\right)
\]

### Step 5: Combine the expressions
Thus, we have two equivalent forms:

\[
\sin(\theta) + \cos(\theta) = x \cdot \cos(\theta) \cdot \left(\frac{1}{x} + \frac{1}{y}\right)
\]

Or:

\[
\sin(\theta) + \cos(\theta) = y \cdot \sin(\theta) \cdot \left(\frac{1}{x} + \frac{1}{y}\right)
\]

These yield equivalent results, depending on the desired variable factor for the final answer. Both approaches give a similar structure for the equation:

The value of \(\sin(\theta) + \cos(\theta)\) in terms of \(x\) and \(y\) is:

\[
\boxed{\frac{1}{\sin(\theta) + \cos(\theta)} = 1}
\] 

This provides a general expression that links the given variables for sine and cosine for the desired result.

### Questions and Tip
1. How does cross-multiplication help in solving trigonometric equations?
2. What other trigonometric identities could simplify expressions like these?
3. Can you express \(\tan(\theta)\) using \(x\) and \(y\) from the given equation?
4. How would the solution change if \(x = y\)?
5. What if instead of sine and cosine, you were given tangent and cotangent?

**Tip:** When dealing with trigonometric identities, always look for ways to simplify by factoring or substituting known identities.

Explore how to determine sin(θ) + cos(θ) in terms of x and y with a detailed solution. This problem involves trigonometric identities and cross-multiplication techniques. Suitable for high school students learning trigonometry.

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