What is the value of $cos(frac{pi}{4})$ on the unit circle?

$$ \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} $$

Solution Steps:

1. Determine the angle and the unit circle: The angle $( \frac{\pi}{4} )$ corresponds to 45° on the unit circle, where the coordinates are $( \cos\left(\frac{\pi}{4}\right), \sin\left(\frac{\pi}{4}\right))$.
2. Use symmetry of the unit circle: At the angle $( \frac{\pi}{4} )$ (45°), the coordinates of the point on the unit circle are equal for both $x$ and $y$. That is:
   $$ \cos\left(\frac{\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right) $$
3. Apply the unit circle identity: According to the unit circle identity, the sum of the squares of the coordinates equals 1:
   $$ \cos^2(\theta) + \sin^2(\theta) = 1 $$
   For $ \theta = \frac{\pi}{4} $, we have:
   $$ \cos^2\left(\frac{\pi}{4}\right) + \cos^2\left(\frac{\pi}{4}\right) = 1 $$
   This simplifies to:
   $$ 2 \cos^2\left(\frac{\pi}{4}\right) = 1 $$
4. Solve for $ \cos\left(\frac{\pi}{4}\right) $: Now, solve the equation:
   $$ \cos^2\left(\frac{\pi}{4}\right) = \frac{1}{2} $$
   Taking the square root of both sides gives:
   $$ \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} $$

$$ \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} $$

Solution Steps:

1. Angle and unit circle: The angle $( \frac{\pi}{4} )$ corresponds to 45°, and the coordinates on the unit circle are $( \cos\left(\frac{\pi}{4}\right), \sin\left(\frac{\pi}{4}\right))$.
2. Symmetry of the unit circle: Since $ \cos\left(\frac{\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right) $, let $x = \cos\left(\frac{\pi}{4}\right)$.
3. Use the unit circle identity: Applying the identity:
   $$ 2x^2 = 1 $$
   Solving for $x$, we get:
   $$ x = \frac{\sqrt{2}}{2} $$

$$ \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} $$

Solution:

The value of $ \cos\left(\frac{\pi}{4}\right) $ is a well-known trigonometric value. From standard trigonometric tables, we know that:

$$ \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} $$

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