To find the sine of 105°, you can use the sine addition formula, which is given by:

\[
\sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B)
\]

We can express 105° as the sum of two angles for which we know the sine and cosine values:

\[
\sin(105°) = \sin(60° + 45°)
\]

Applying the sine addition formula:

\[
\sin(105°) = \sin(60°)\cos(45°) + \cos(60°)\sin(45°)
\]

Now, using the known values:

\[
\sin(60°) = \frac{\sqrt{3}}{2}, \quad \cos(45°) = \frac{\sqrt{2}}{2}, \quad \cos(60°) = \frac{1}{2}, \quad \sin(45°) = \frac{\sqrt{2}}{2}
\]

Substituting these into the formula:

\[
\sin(105°) = \left(\frac{\sqrt{3}}{2}\right) \left(\frac{\sqrt{2}}{2}\right) + \left(\frac{1}{2}\right) \left(\frac{\sqrt{2}}{2}\right)
\]

\[
\sin(105°) = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}
\]

\[
\sin(105°) = \frac{\sqrt{6} + \sqrt{2}}{4}
\]

So, the exact value of \(\sin(105°)\) is \(\frac{\sqrt{6} + \sqrt{2}}{4}\).

Would you like more details or have any questions?

### Follow-up Questions:

1. What is the value of \(\cos(105°)\)?
2. How do you use the sine subtraction formula?
3. Can you calculate \(\sin(75°)\) using a similar method?
4. How can you find the sine of other angles that are not standard angles?
5. What is the difference between sine and cosine functions?
6. How do you solve trigonometric equations involving sine?
7. Can you explain how to use a calculator to find \(\sin(105°)\)?
8. What are the applications of sine in real-world problems?

### Tip:
When dealing with non-standard angles, using angle sum or difference identities is a helpful strategy to find exact trigonometric values.

Learn how to calculate the sine of 105 degrees using trigonometric identities and the sine addition formula. This detailed solution includes step-by-step calculations and formula application. Suitable for high school students studying trigonometry.

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