To find the sine of $45^\circ$ using the unit circle, follow these steps:

1. Locate $45^\circ$ on the Unit Circle:

   In the unit circle, $45^\circ$ is in the first quadrant. The coordinates of the point on the unit circle corresponding to $45^\circ$ are $\left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right)$.

2. Find the Sine Value:

   The sine of an angle in the unit circle is given by the y-coordinate of the corresponding point. Thus, for $45^\circ$:

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

Summary

• The green dot should be located at the point $\left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right)$ on the unit circle.
• The exact value of $\sin 45^\circ$ is $\frac{\sqrt{2}}{2}$.

Would you like more details or have any questions?

Here are 8 related questions to expand on this topic:

1. What is the cosine of $45^\circ$?
2. How can you derive the exact values of sine and cosine for $45^\circ$ using right triangles?
3. What are the coordinates of the point on the unit circle corresponding to $90^\circ$?
4. How do you determine the sine of $30^\circ$?
5. What is the tangent of $45^\circ$?
6. How can you find the sine of $60^\circ$ using the unit circle?
7. What is the relationship between the sine and cosine of complementary angles?
8. How does the unit circle help in understanding trigonometric identities?

Tip: To easily remember the exact values of sine and cosine for commonly used angles like $30^\circ$, $45^\circ$, and $60^\circ$, it can be helpful to visualize the unit circle and the corresponding right triangles.