What is the value of sin `292(1)/(2)""^@`?

Options

A. `1/3sqrt(2+sqrt3)`
B. `-1/3sqrt(2-sqrt3)`
C. `1/2sqrt(2+sqrt2)`
D. `-1/2sqrt(2+sqrt2)`

Answer

To find the value of \( \sin 292.5^\circ \), we can use the half-angle formula for sine. Here’s a step-by-step solution:

### Step 1: Identify the angle
We need to calculate \( \sin 292.5^\circ \).

### Step 2: Use the half-angle formula
The half-angle formula for sine is given by:
\[ \sin \theta = \sqrt{\frac{1 - \cos 2\theta}{2}} \]
In our case, we can express \( 292.5^\circ \) as \( \frac{585^\circ}{2} \). Thus, we can set \( \theta = 292.5^\circ \) and \( 2\theta = 585^\circ \).

### Step 3: Substitute into the formula
Using the half-angle formula:
\[ \sin 292.5^\circ = \sqrt{\frac{1 - \cos 585^\circ}{2}} \]

### Step 4: Simplify \( \cos 585^\circ \)
To simplify \( \cos 585^\circ \), we can reduce it using the periodic property of cosine:
\[ \cos 585^\circ = \cos(585^\circ - 540^\circ) = \cos 45^\circ \]
Since \( 585^\circ \) is \( 540^\circ + 45^\circ \) and \( \cos(540^\circ + \theta) = -\cos \theta \):
\[ \cos 585^\circ = -\cos 45^\circ = -\frac{1}{\sqrt{2}} \]

### Step 5: Substitute back into the sine formula
Now we substitute \( \cos 585^\circ \) back into our sine formula:
\[ \sin 292.5^\circ = \sqrt{\frac{1 - (-\frac{1}{\sqrt{2}})}{2}} = \sqrt{\frac{1 + \frac{1}{\sqrt{2}}}{2}} \]

### Step 6: Simplify the expression
To simplify \( 1 + \frac{1}{\sqrt{2}} \):
\[ 1 + \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{\sqrt{2}} + \frac{1}{\sqrt{2}} = \frac{\sqrt{2} + 1}{\sqrt{2}} \]
Thus,
\[ \sin 292.5^\circ = \sqrt{\frac{\frac{\sqrt{2} + 1}{\sqrt{2}}}{2}} = \sqrt{\frac{\sqrt{2} + 1}{2\sqrt{2}}} \]

### Step 7: Rationalize the denominator
To rationalize the denominator:
\[ \sin 292.5^\circ = \sqrt{\frac{\sqrt{2} + 1}{2\sqrt{2}}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \sqrt{\frac{(\sqrt{2} + 1)\sqrt{2}}{4}} = \frac{\sqrt{2(\sqrt{2} + 1)}}{2} \]

### Step 8: Final simplification
This can be simplified further:
\[ \sin 292.5^\circ = \frac{\sqrt{2 + 2\sqrt{2}}}{2} \]

### Conclusion
Thus, the value of \( \sin 292.5^\circ \) is:
\[ \sin 292.5^\circ = \frac{1}{2} \sqrt{2 + 2\sqrt{2}} \]