Find the value of $cos(frac{pi}{9})$ using the unit circle and trigonometric identities

To find the value of $\cos(\frac{\pi}{9})$, we can utilize the triple angle formula for cosine: $\cos(3\theta) = 4\cos^3(\theta) – 3\cos(\theta)$. Let $\theta = \frac{\pi}{9}$.

Therefore, $3\theta = \frac{3\pi}{9} = \frac{\pi}{3}$, and we know that $\cos(\frac{\pi}{3}) = \frac{1}{2}$.

Substituting these values into the triple angle formula, we get:

$\cos(\frac{\pi}{3}) = 4\cos^3(\frac{\pi}{9}) – 3\cos(\frac{\pi}{9})$

$\frac{1}{2} = 4\cos^3(\frac{\pi}{9}) – 3\cos(\frac{\pi}{9})$

Let $x = \cos(\frac{\pi}{9})$, then we have the cubic equation:

$\frac{1}{2} = 4x^3 – 3x$

Rearranging gives:

$4x^3 – 3x – \frac{1}{2} = 0$

Using numerical methods, the solution is:

$\cos(\frac{\pi}{9}) \approx 0.9848$

To find $cos(frac{pi}{9})$ using the unit circle, we employ angle subtraction and trigonometric identities. Consider $ heta = frac{pi}{9}$. We use the identity $cos(3 heta) = 4cos^3( heta) – 3cos( heta)$.

Since $3 heta = frac{pi}{3}$, and $cos(frac{pi}{3}) = frac{1}{2}$, we write:

$cos(frac{pi}{3}) = 4cos^3(frac{pi}{9}) – 3cos(frac{pi}{9})$

Let $x = cos(frac{pi}{9})$. Thus:

$4x^3 – 3x = frac{1}{2}$

Rearranging yields:

$8x^3 – 6x – 1 = 0$

Solving this cubic equation using algebraic methods or numerical approximation:

$cos(frac{pi}{9}) approx 0.9848$

To determine $cos(frac{pi}{9})$, use the triple angle formula: $cos(3 heta) = 4cos^3( heta) – 3cos( heta)$. With $ heta = frac{pi}{9}$:

$cos(frac{pi}{3}) = 4cos^3(frac{pi}{9}) – 3cos(frac{pi}{9})$

Let $x = cos(frac{pi}{9})$:

$4x^3 – 3x = frac{1}{2}$

Solve for $x$:

$8x^3 – 6x – 1 = 0$

Approximate solution:

$cos(frac{pi}{9}) approx 0.9848$

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