Given : `sin theta = p/q` , find `cos theta + sin theta ` in terms of p and q.

Answer

To solve the problem step by step, we will start with the given information and proceed to find \( \cos \theta + \sin \theta \) in terms of \( p \) and \( q \).

### Step 1: Write down the given information
We are given:
\[ \sin \theta = \frac{p}{q} \]

### Step 2: Use the Pythagorean identity
We know from trigonometric identities that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substituting the value of \( \sin \theta \):
\[ \left(\frac{p}{q}\right)^2 + \cos^2 \theta = 1 \]

### Step 3: Solve for \( \cos^2 \theta \)
Rearranging the equation gives:
\[ \cos^2 \theta = 1 - \left(\frac{p}{q}\right)^2 \]
Calculating \( \left(\frac{p}{q}\right)^2 \):
\[ \cos^2 \theta = 1 - \frac{p^2}{q^2} \]
To combine the terms, we express 1 as \( \frac{q^2}{q^2} \):
\[ \cos^2 \theta = \frac{q^2 - p^2}{q^2} \]

### Step 4: Take the square root to find \( \cos \theta \)
Taking the square root of both sides, we have:
\[ \cos \theta = \sqrt{\frac{q^2 - p^2}{q^2}} = \frac{\sqrt{q^2 - p^2}}{q} \]

### Step 5: Write the expression \( \cos \theta + \sin \theta \)
Now we need to find \( \cos \theta + \sin \theta \):
\[ \cos \theta + \sin \theta = \frac{\sqrt{q^2 - p^2}}{q} + \frac{p}{q} \]

### Step 6: Combine the terms
Combining the two fractions over a common denominator:
\[ \cos \theta + \sin \theta = \frac{\sqrt{q^2 - p^2} + p}{q} \]

### Final Answer
Thus, the expression for \( \cos \theta + \sin \theta \) in terms of \( p \) and \( q \) is:
\[ \cos \theta + \sin \theta = \frac{p + \sqrt{q^2 - p^2}}{q} \]

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