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!