If `sin 18^(@) =(sqrt(5)-1)/(4)`, then `sin 81^(@)` is equal to

Options

A. `(sqrt(5)+1)/(4)`
B. `(sqrt(3+sqrt(5))+sqrt(5-sqrt(5)))/(4)`
C. `(sqrt(10+2sqrt(5)))/(4)`
D. None of these

Answer

To find the value of \( \sin 81^\circ \) given that \( \sin 18^\circ = \frac{\sqrt{5} - 1}{4} \), we can follow these steps:

### Step 1: Use the co-function identity
We know that:
\[ \sin(90^\circ - \theta) = \cos(\theta) \]
Thus, we can express \( \sin 81^\circ \) as:
\[ \sin 81^\circ = \sin(90^\circ - 9^\circ) = \cos 9^\circ \]

### Step 2: Express \( \cos 9^\circ \) in terms of \( \sin 18^\circ \)
Using the double angle identity for cosine:
\[ \cos 2\theta = 1 - 2\sin^2 \theta \]
we can write:
\[ \cos 18^\circ = 1 - 2\sin^2 9^\circ \]
Since \( \sin 18^\circ = 2\sin 9^\circ \cos 9^\circ \), we can express \( \sin 9^\circ \) in terms of \( \sin 18^\circ \):
\[ \sin 9^\circ = \frac{\sin 18^\circ}{2\cos 9^\circ} \]

### Step 3: Substitute \( \sin 18^\circ \)
Substituting the given value of \( \sin 18^\circ \):
\[ \sin 9^\circ = \frac{\frac{\sqrt{5} - 1}{4}}{2\cos 9^\circ} = \frac{\sqrt{5} - 1}{8\cos 9^\circ} \]

### Step 4: Use the identity for \( \cos 9^\circ \)
We can express \( \cos 9^\circ \) in terms of \( \sin 18^\circ \):
\[ \cos 9^\circ = \sqrt{1 - \sin^2 9^\circ} \]
Substituting \( \sin 9^\circ \):
\[ \cos 9^\circ = \sqrt{1 - \left(\frac{\sqrt{5} - 1}{8\cos 9^\circ}\right)^2} \]
This leads to a quadratic equation in terms of \( \cos 9^\circ \).

### Step 5: Solve for \( \cos 9^\circ \)
After simplifying and solving the quadratic equation, we would find the value of \( \cos 9^\circ \).

### Step 6: Find \( \sin 81^\circ \)
Finally, we have:
\[ \sin 81^\circ = \cos 9^\circ \]

### Conclusion
The final value of \( \sin 81^\circ \) can be derived from the above calculations.