If a cos θ – b sin θ = c, then a sin θ + b cos θ =

Question

Accepted Answer

To solve the problem and justify why Option 2 is correct, we need to analyze the given equation and derive the expression for $ a \sin \theta + b \cos \theta $.

Given:
$$ a \cos \theta - b \sin \theta = c $$

We need to find:
$$ a \sin \theta + b \cos \theta $$

To solve this, we can use the identity for the sum of squares of trigonometric expressions. Consider squaring both sides of the given equation:

$$ (a \cos \theta - b \sin \theta)^2 = c^2 $$

Expanding the left side, we get:
$$ a^2 \cos^2 \theta - 2ab \cos \theta \sin \theta + b^2 \sin^2 \theta = c^2 $$

Using the Pythagorean identity $\cos^2 \theta + \sin^2 \theta = 1$, we can rewrite the equation as:
$$ a^2 (1 - \sin^2 \theta) + b^2 \sin^2 \theta - 2ab \cos \theta \sin \theta = c^2 $$

Simplifying, we have:
$$ a^2 - a^2 \sin^2 \theta + b^2 \sin^2 \theta - 2ab \cos \theta \sin \theta = c^2 $$

Rearranging terms, we get:
$$ a^2 + (b^2 - a^2) \sin^2 \theta - 2ab \cos \theta \sin \theta = c^2 $$

Now, let's consider the expression $ a \sin \theta + b \cos \theta $ and square it:
$$ (a \sin \theta + b \cos \theta)^2 = a^2 \sin^2 \theta + 2ab \sin \theta \cos \theta + b^2 \cos^2 \theta $$

Using the identity $\cos^2 \theta + \sin^2 \theta = 1$, we can rewrite this as:
$$ a^2 \sin^2 \theta + 2ab \sin \theta \cos \theta + b^2 (1 - \sin^2 \theta) $$

Simplifying, we have:
$$ a^2 \sin^2 \theta + 2ab \sin \theta \cos \theta + b^2 - b^2 \sin^2 \theta $$

Rearranging terms, we get:
$$ (a^2 - b^2) \sin^2 \theta + 2ab \sin \theta \cos \theta + b^2 $$

From the earlier equation derived from the given condition, we know:
$$ a^2 + (b^2 - a^2) \sin^2 \theta - 2ab \cos \theta \sin \theta = c^2 $$

By comparing these expressions, we can see that:
$$ (a \sin \theta + b \cos \theta)^2 = a^2 + b^2 - c^2 $$

Thus, taking the square root of both sides, we find:
$$ a \sin \theta + b \cos \theta = \pm \sqrt{a^2 + b^2 - c^2} $$

Therefore, the correct answer is Option 2: $\pm \sqrt{a^2 + b^2 - c^2}$.

**Explanation for Incorrect Options:**

1. **Option 1: $\pm \sqrt{a^2 + b^2 + c^2}$**
   - This option incorrectly adds $c^2$ instead of subtracting it. The derivation shows that the correct expression involves subtracting $c^2$.

3. **Option 3: $\pm \sqrt{c^2 - a^2 - b^2}$**
   - This option incorrectly rearranges the terms, placing $c^2$ as the leading term and subtracting $a^2$ and $b^2$, which does not match the derived expression.

4. **Option 4: $\pm \sqrt{b^2 - a^2 - c^2}$**
   - This option incorrectly subtracts both $a^2$ and $c^2$ from $b^2$, which does not align with the correct derivation.

In conclusion, the correct answer is Option 2: $\pm \sqrt{a^2 + b^2 - c^2}$, as derived from the given equation and verified through logical reasoning.