Question

If \(\sin \theta - \cos \theta = \frac{4}{5}\) then the value of \(\sin \theta + \cos \theta =\)

• A. \(\frac{5}{\sqrt{34}}\)
• B. \(-\frac{5}{\sqrt{34}}\)
• C. \(\frac{\sqrt{34}}{25}\)
• D. \(\frac{\sqrt{34}}{5}\)

Hint:

The identity \((\sin\theta \pm \cos\theta)^2 = 1 \pm 2\sin\theta\cos\theta\) is extremely useful. Memorizing the combined identity \((\sin\theta - \cos\theta)^2 + (\sin\theta + \cos\theta)^2 = 2\) provides a direct shortcut for problems where one expression is given and the other is asked.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We are given the value of \(\sin \theta - \cos \theta\) and we need to find the value of \(\sin \theta + \cos \theta\).

Step 2: Key Formula or Approach:
This is a standard problem type that uses the relationship between \((\sin \theta - \cos \theta)^2\) and \((\sin \theta + \cos \theta)^2\).
We know that:
\((\sin \theta - \cos \theta)^2 = \sin^2 \theta - 2\sin \theta \cos \theta + \cos^2 \theta = 1 - 2\sin \theta \cos \theta\)
\((\sin \theta + \cos \theta)^2 = \sin^2 \theta + 2\sin \theta \cos \theta + \cos^2 \theta = 1 + 2\sin \theta \cos \theta\)
By adding these two equations, we get a direct relationship:
\((\sin \theta - \cos \theta)^2 + (\sin \theta + \cos \theta)^2 = 2\)

Step 3: Detailed Explanation:
Let \(x = \sin \theta + \cos \theta\). We need to find the value of \(x\).
We are given \(\sin \theta - \cos \theta = \frac{4}{5}\).
Using the identity from Step 2:
\[ (\sin \theta - \cos \theta)^2 + (\sin \theta + \cos \theta)^2 = 2 \]
Substitute the given values into this identity:
\[ \left(\frac{4}{5}\right)^2 + x^2 = 2 \]
\[ \frac{16}{25} + x^2 = 2 \]
Now, solve for \(x^2\):
\[ x^2 = 2 - \frac{16}{25} \]
\[ x^2 = \frac{50 - 16}{25} \]
\[ x^2 = \frac{34}{25} \]
Take the square root of both sides:
\[ x = \pm \sqrt{\frac{34}{25}} = \pm \frac{\sqrt{34}}{5} \]
Since one of the options is \(\frac{\sqrt{34}}{5}\), we select this value. The question doesn't provide enough information to determine the sign, but only the positive value is listed as a primary option choice.

Step 4: Final Answer:
The value of \(\sin \theta + \cos \theta\) is \(\frac{\sqrt{34}}{5}\).