Find the sine and cosine values for the angle $ heta = 45^{circ}$ on the unit circle.

To find the sine and cosine values for the angle $\theta = 45^{\circ}$ on the unit circle:

1. Note that $\theta = 45^{\circ}$ is in the first quadrant.

2. The coordinates of the corresponding point on the unit circle are given by $(\cos(45^{\circ}), \sin(45^{\circ}))$.

3. Using standard values:

$\cos(45^{\circ}) = \frac{\sqrt{2}}{2}$

$\sin(45^{\circ}) = \frac{\sqrt{2}}{2}$

Thus, the sine and cosine values for $\theta = 45^{\circ}$ are both $\frac{\sqrt{2}}{2}$.

To determine the sine and cosine for $ heta = 45^{circ}$ on the unit circle:

1. Recognize that $ heta = 45^{circ}$ is located in the first quadrant.

2. The coordinates on the unit circle for $ heta = 45^{circ}$ are $(cos(45^{circ}), sin(45^{circ}))$.

3. Recall that:

$cos(45^{circ}) = sin(45^{circ}) = frac{sqrt{2}}{2}$

Thus, the values are $cos(45^{circ}) = frac{sqrt{2}}{2}$ and $sin(45^{circ}) = frac{sqrt{2}}{2}$.

For $ heta = 45^{circ}$, the unit circle coordinates are:

$cos(45^{circ}) = frac{sqrt{2}}{2}$

$sin(45^{circ}) = frac{sqrt{2}}{2}$

Thus, both $cos(45^{circ})$ and $sin(45^{circ})$ are equal to $frac{sqrt{2}}{2}$.

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