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The image presents a mathematical statement regarding periodic functions. The problem seems to be asking for definitions related to periodic functions and trigonometric functions like sine and cosine.

Here’s a breakdown of how to complete the blanks:

1. **First blank (related to \( f(t + p) \)):**
   - For a periodic function \( f \) with period \( p \), we have the property:
   \[
   f(t + p) = f(t) \quad \text{for all} \, t
   \]
   So, the correct answer to fill this blank is \( f(t) \).

2. **Second blank (period of sine and cosine functions):**
   - The sine and cosine functions, \( y = \sin(x) \) and \( y = \cos(x) \), are periodic with a period of \( 2\pi \). This is because these functions repeat every \( 2\pi \) radians.

3. **Third blank (amplitude of sine and cosine functions):**
   - The amplitude of both \( \sin(x) \) and \( \cos(x) \) is 1, since their maximum value is 1 and their minimum value is -1.

### Final Complete Statement:
- "If a function \( f \) is periodic with period \( p \), then \( f(t + p) = f(t) \) for every \( t \). The trigonometric functions \( y = \sin(x) \) and \( y = \cos(x) \) are periodic, with period \( 2\pi \) and amplitude 1."

Would you like further details on periodic functions, or do you have any questions?

Here are 5 related questions to deepen your understanding:

1. What does it mean for a function to have a period?
2. How can you determine the period of a more complex function, such as \( y = \sin(2x) \)?
3. What is the difference between the amplitude and the period of a trigonometric function?
4. How does the period of a function change when you modify the input (e.g., \( \sin(kx) \))?
5. Can you think of examples of periodic functions outside of trigonometry?

**Tip:** When determining the period of a function, always check the interval after which the function repeats its values. For trigonometric functions, this interval is typically based on \( 2\pi \).

If a function f is periodic with period p, then f(t + p) = ___ for every t. The trigonometric functions y = sin(x) and y = cos(x) are periodic, with period ___ and amplitude ___.

Explore periodic functions and understand the periodicity and amplitude of sine and cosine functions. This math problem involves defining the period of periodic functions and calculating amplitudes, making it ideal for students in Grades 9-12.

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