What is the value of `theta, 0^(@) lt theta lt 90^(@), " if " cos theta + sin theta = sqrt(2)`?

Options

A. `30^(@)`
B. `60^(@)`
C. `15^(@)`
D. `45^(@)`

Answer

To solve the equation \( \cos \theta + \sin \theta = \sqrt{2} \) for \( 0^\circ < \theta < 90^\circ \), we can follow these steps:

### Step 1: Square both sides of the equation
Starting with the equation:
\[ \cos \theta + \sin \theta = \sqrt{2} \]
we square both sides:
\[ (\cos \theta + \sin \theta)^2 = (\sqrt{2})^2 \]

### Step 2: Expand the left side
Using the identity \( (a + b)^2 = a^2 + 2ab + b^2 \), we expand the left side:
\[ \cos^2 \theta + 2 \cos \theta \sin \theta + \sin^2 \theta = 2 \]

### Step 3: Use the Pythagorean identity
We know from the Pythagorean identity that \( \cos^2 \theta + \sin^2 \theta = 1 \). Substituting this into the equation gives:
\[ 1 + 2 \cos \theta \sin \theta = 2 \]

### Step 4: Simplify the equation
Now, we can simplify the equation:
\[ 2 \cos \theta \sin \theta = 2 - 1 \]
\[ 2 \cos \theta \sin \theta = 1 \]

### Step 5: Use the double angle identity
We can use the double angle identity for sine, which states that \( \sin 2\theta = 2 \sin \theta \cos \theta \):
\[ \sin 2\theta = 1 \]

### Step 6: Solve for \( 2\theta \)
The equation \( \sin 2\theta = 1 \) implies:
\[ 2\theta = 90^\circ \]

### Step 7: Solve for \( \theta \)
Dividing both sides by 2 gives:
\[ \theta = 45^\circ \]

### Conclusion
Thus, the value of \( \theta \) is:
\[ \theta = 45^\circ \]