Given the following unit circle, rotate green dot to the appropriate angle and then find the exact value of the function.
sine, 45, degrees
sin45
∘

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.

Learn how to find the exact value of \( \sin 45^\circ \) using the unit circle. Detailed steps and explanation provided, including the coordinates of the point on the unit circle corresponding to \( 45^\circ \) and the resulting sine value.

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