To solve $\sin 75^\circ$ using the sum or difference formula, let's break it down step by step:

Formula for $\sin(A + B)$:

$\sin(A + B) = \sin A \cos B + \cos A \sin B$

Choosing $A$ and $B$:

We can express $75^\circ$ as $45^\circ + 30^\circ$. Therefore: $\sin 75^\circ = \sin(45^\circ + 30^\circ)$

Substituting into the formula:

$\sin(45^\circ + 30^\circ) = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ$

Using the known values of trigonometric functions:

• $\sin 45^\circ = \cos 45^\circ = \frac{\sqrt{2}}{2}$,
• $\sin 30^\circ = \frac{1}{2}$,
• $\cos 30^\circ = \frac{\sqrt{3}}{2}$.

Substitute these values: $\sin 75^\circ = \left(\frac{\sqrt{2}}{2}\right) \left(\frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2}\right) \left(\frac{1}{2}\right)$

Simplify:

$\sin 75^\circ = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}$

Combine the terms: $\sin 75^\circ = \frac{\sqrt{6} + \sqrt{2}}{4}$

Final Answer:

$\sin 75^\circ = \frac{\sqrt{6} + \sqrt{2}}{4}$

Let me know if you'd like further clarification or a breakdown of the steps!

Here are 5 related questions to expand your understanding:

1. How would you calculate $\cos 75^\circ$ using a similar approach?
2. Can the same formula be used to find $\sin 15^\circ$?
3. What are the exact values of $\tan 75^\circ$ using $\sin 75^\circ$ and $\cos 75^\circ$?
4. Why is it useful to memorize exact trigonometric values for common angles?
5. How does the sum or difference formula apply to radians instead of degrees?

Tip: Memorizing the unit circle values for common angles like $30^\circ$, $45^\circ$, and $60^\circ$ will save you time when solving trigonometric problems!