What is the value of cos `36^@`?

Options

A. `(sqrt5-1)/(4)`
B. `(sqrt5+1)/(4)`
C. `(sqrt(10+22sqrt5))/(4)`
D. `(sqrt(10-2sqrt5))/(4)`

Answer

To find the value of \( \cos 36^\circ \), we can follow these steps:

### Step 1: Find \( \sin 18^\circ \)

Let \( x = 18^\circ \). We know that \( 5x = 90^\circ \). We can express \( 5x \) as \( 2x + 3x \).

### Step 2: Use the sine identity

Using the identity \( \sin(90^\circ - \theta) = \cos(\theta) \), we have:
\[ \sin 2x = \sin(90^\circ - 3x) = \cos 3x \]

### Step 3: Apply the sine double angle formula

Using the double angle formula, we can write:
\[ \sin 2x = 2 \sin x \cos x \]
Thus, we have:
\[ 2 \sin x \cos x = \cos 3x \]

### Step 4: Apply the cosine triple angle formula

Using the triple angle formula, we know:
\[ \cos 3x = 4 \cos^3 x - 3 \cos x \]

### Step 5: Set up the equation

From the previous steps, we can set up the equation:
\[ 2 \sin x \cos x = 4 \cos^3 x - 3 \cos x \]
Rearranging gives us:
\[ 2 \sin x = 4 \cos^2 x - 3 \]

### Step 6: Substitute \( \sin^2 x \)

Using the identity \( \sin^2 x = 1 - \cos^2 x \), we can substitute:
\[ 2 \sqrt{1 - \cos^2 x} = 4 \cos^2 x - 3 \]

### Step 7: Square both sides

Squaring both sides results in:
\[ 4(1 - \cos^2 x) = (4 \cos^2 x - 3)^2 \]

### Step 8: Expand and simplify

Expanding the right-hand side:
\[ 4 - 4 \cos^2 x = 16 \cos^4 x - 24 \cos^2 x + 9 \]
Rearranging gives:
\[ 16 \cos^4 x - 20 \cos^2 x + 5 = 0 \]

### Step 9: Solve the quadratic equation

Let \( y = \cos^2 x \). The equation becomes:
\[ 16y^2 - 20y + 5 = 0 \]
Using the quadratic formula:
\[ y = \frac{-(-20) \pm \sqrt{(-20)^2 - 4 \cdot 16 \cdot 5}}{2 \cdot 16} \]
Calculating the discriminant:
\[ 400 - 320 = 80 \]
Thus:
\[ y = \frac{20 \pm \sqrt{80}}{32} = \frac{20 \pm 4\sqrt{5}}{32} = \frac{5 \pm \sqrt{5}}{8} \]

### Step 10: Find \( \cos 36^\circ \)

Since \( \cos 36^\circ = \sqrt{y} \), we take the positive root:
\[ \cos 36^\circ = \sqrt{\frac{5 + \sqrt{5}}{8}} = \frac{\sqrt{5} + 1}{4} \]

### Final Answer

Thus, the value of \( \cos 36^\circ \) is:
\[ \cos 36^\circ = \frac{\sqrt{5} + 1}{4} \]

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