The value of `sin18^(@)` is

Options

A. `(sqrt(5)+1)/(4)`
B. `(sqrt(5)-1)/(4)`
C. `(4)/(sqrt(5)+1)`
D. `(4)/(sqrt(5)-1)`

Explanation

To find the value of \( \sin 18^\circ \), we can follow these steps:

### Step 1: Define the angle
Let \( \theta = 18^\circ \). Therefore, \( 2\theta = 36^\circ \).

### Step 2: Use the double angle formula
Using the double angle formula for sine, we have:
\[ \sin 2\theta = 2 \sin \theta \cos \theta \]
This means:
\[ \sin 36^\circ = 2 \sin 18^\circ \cos 18^\circ \]

### Step 3: Relate \( \sin 36^\circ \) to \( \cos 54^\circ \)
We know that:
\[ \sin 36^\circ = \cos 54^\circ \]
And since \( 54^\circ = 90^\circ - 36^\circ \), we can write:
\[ \sin 36^\circ = \sin(90^\circ - 54^\circ) = \cos 54^\circ \]

### Step 4: Use the identity for \( \cos 54^\circ \)
Using the identity \( \cos 54^\circ = \sin 36^\circ \), we can express:
\[ \sin 36^\circ = \cos 54^\circ = \sin(90^\circ - 36^\circ) = \sin 36^\circ \]

### Step 5: Set up the equation
Now substituting back, we have:
\[ \sin 36^\circ = 2 \sin 18^\circ \cos 18^\circ \]
Substituting \( \sin 36^\circ \) gives us:
\[ \cos 54^\circ = 2 \sin 18^\circ \cos 18^\circ \]

### Step 6: Use the cosine double angle formula
We can also express \( \cos 54^\circ \) using the double angle formula:
\[ \cos 54^\circ = 2 \cos^2 27^\circ - 1 \]

### Step 7: Solve the quadratic equation
Now we can rearrange the equation:
\[ 2 \sin 18^\circ \cos 18^\circ - \cos 54^\circ = 0 \]
This leads us to a quadratic equation in terms of \( \sin \theta \):
\[ -4 \sin^2 \theta - 2 \sin \theta + 1 = 0 \]

### Step 8: Use the quadratic formula
Using the quadratic formula:
\[ \sin \theta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = -4, b = -2, c = 1 \):
\[ \sin 18^\circ = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot (-4) \cdot 1}}{2 \cdot (-4)} \]
\[ = \frac{2 \pm \sqrt{4 + 16}}{-8} \]
\[ = \frac{2 \pm \sqrt{20}}{-8} \]
\[ = \frac{2 \pm 2\sqrt{5}}{-8} \]
\[ = \frac{-1 \pm \sqrt{5}}{4} \]

### Step 9: Choose the positive root
Since \( \sin 18^\circ \) is positive in the first quadrant:
\[ \sin 18^\circ = \frac{\sqrt{5} - 1}{4} \]

### Final Answer
Thus, the value of \( \sin 18^\circ \) is:
\[ \sin 18^\circ = \frac{\sqrt{5} - 1}{4} \]

---