Emily has been asked to create a game using cupcakes for a convention. The organizers have given her an interesting challenge: they want her to decorate cupcakes that incorporate math problems! Specifically, they would like her to decorate cupcakes that use trigonometry formulas. Emily decides to create cupcakes that involve the trigonometric identities where the event attendees will be challenged to match up the expression on one cupcake with the correct rewritten answer displayed on another cupcake. 
 
One of the problems she needs to solve involves rewriting sin(x+5π6)sinx+5π6 in terms of sinxsinx and cosxcosx. How  would Emily rewrite this for the matching cupcake in the game? 
 
Enclose arguments of functions in parentheses. For example, sin(2x)sin2x.
 
sin(x+5π6)=

To solve the problem where Emily needs to rewrite the expression \( \sin\left(x + \frac{5\pi}{6}\right) \) in terms of \( \sin(x) \) and \( \cos(x) \), we can use the angle sum identity for sine.

### Angle Sum Identity:
\[
\sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B)
\]

In this case:
- \( A = x \)
- \( B = \frac{5\pi}{6} \)

So, applying the identity:
\[
\sin\left(x + \frac{5\pi}{6}\right) = \sin(x)\cos\left(\frac{5\pi}{6}\right) + \cos(x)\sin\left(\frac{5\pi}{6}\right)
\]

### Now, let's find the values of \( \cos\left(\frac{5\pi}{6}\right) \) and \( \sin\left(\frac{5\pi}{6}\right) \):

1. \( \cos\left(\frac{5\pi}{6}\right) = -\cos\left(\frac{\pi}{6}\right) = -\frac{\sqrt{3}}{2} \)
2. \( \sin\left(\frac{5\pi}{6}\right) = \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \)

### Substitute these into the expression:
\[
\sin\left(x + \frac{5\pi}{6}\right) = \sin(x) \cdot \left(-\frac{\sqrt{3}}{2}\right) + \cos(x) \cdot \frac{1}{2}
\]

This simplifies to:
\[
\sin\left(x + \frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2} \sin(x) + \frac{1}{2} \cos(x)
\]

### Final Answer:
\[
\sin\left(x + \frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2} \sin(x) + \frac{1}{2} \cos(x)
\]

Emily would decorate the matching cupcake with this expression. 

Would you like further details or explanations? Here are some additional related questions you might explore:

1. What is the angle sum identity for cosine, and how can it be applied to similar problems?
2. How can trigonometric identities be used to simplify complex expressions?
3. What are the values of sine and cosine for other special angles like \( \frac{\pi}{4} \) and \( \frac{\pi}{3} \)?
4. How do trigonometric identities relate to the unit circle?
5. What are some real-world applications of trigonometric identities beyond mathematics?

**Tip:** When solving problems involving trigonometric identities, always start by considering if an angle sum or difference identity can be applied, as it often simplifies the expression significantly.

Learn how to rewrite sin(x + 5π/6) in terms of sin(x) and cos(x) with a detailed step-by-step solution. This problem involves applying the angle sum identity for sine, providing insights into trigonometric identities. Suitable for advanced high school students.

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