To solve the expression \(\sin 15^\circ + \cos 15^\circ\), let's compute the exact values for both \(\sin 15^\circ\) and \(\cos 15^\circ\).

### Step-by-Step Calculation

1. **Using the Half-Angle Formulas:**

   To find \(\sin 15^\circ\) and \(\cos 15^\circ\), note that \(15^\circ = 45^\circ - 30^\circ\). However, another approach is using the half-angle formulas for trigonometric functions because:

   \[
   15^\circ = \frac{30^\circ}{2}
   \]

   For a half-angle \(\frac{\theta}{2}\), we have:

   \[
   \sin \frac{\theta}{2} = \sqrt{\frac{1 - \cos \theta}{2}}, \quad \cos \frac{\theta}{2} = \sqrt{\frac{1 + \cos \theta}{2}}
   \]

2. **Calculate \(\sin 15^\circ\) and \(\cos 15^\circ\):**

   Let's take \(\theta = 30^\circ\):

   - \(\cos 30^\circ = \frac{\sqrt{3}}{2}\)

   Now, apply the half-angle formulas:

   \[
   \sin 15^\circ = \sin \frac{30^\circ}{2} = \sqrt{\frac{1 - \cos 30^\circ}{2}} = \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}} = \sqrt{\frac{2 - \sqrt{3}}{4}} = \frac{\sqrt{2 - \sqrt{3}}}{2}
   \]

   \[
   \cos 15^\circ = \cos \frac{30^\circ}{2} = \sqrt{\frac{1 + \cos 30^\circ}{2}} = \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}} = \sqrt{\frac{2 + \sqrt{3}}{4}} = \frac{\sqrt{2 + \sqrt{3}}}{2}
   \]

3. **Add \(\sin 15^\circ\) and \(\cos 15^\circ\):**

   Now we add these two expressions:

   \[
   \sin 15^\circ + \cos 15^\circ = \frac{\sqrt{2 - \sqrt{3}}}{2} + \frac{\sqrt{2 + \sqrt{3}}}{2}
   \]

   Combine the fractions:

   \[
   \sin 15^\circ + \cos 15^\circ = \frac{\sqrt{2 - \sqrt{3}} + \sqrt{2 + \sqrt{3}}}{2}
   \]

This is the simplified exact value for the expression \(\sin 15^\circ + \cos 15^\circ\).

### Summary

\[
\sin 15^\circ + \cos 15^\circ = \frac{\sqrt{2 - \sqrt{3}} + \sqrt{2 + \sqrt{3}}}{2}
\]

Would you like more details on this calculation or have any questions?

### Follow-Up Questions

1. What is the geometric interpretation of \(\sin 15^\circ\) and \(\cos 15^\circ\) on the unit circle?
2. How can we use other trigonometric identities to verify \(\sin 15^\circ + \cos 15^\circ\)?
3. Can we express \(\sin 15^\circ + \cos 15^\circ\) in terms of other angles?
4. How would the calculation change for \(\sin 75^\circ + \cos 75^\circ\)?
5. What are other ways to derive the exact values of \(\sin 15^\circ\) and \(\cos 15^\circ\) using angle difference identities?

### Tip

Remember that trigonometric identities can often simplify expressions and calculations involving angles!

Explore how to calculate sin 15 degrees + cos 15 degrees with a detailed step-by-step solution. Learn the half-angle formulas and trigonometric identities involved. Suitable for high school students studying trigonometry.

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