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The problem asks us to determine whether the following statement is true or false:

The graph of \( y = \cos \left( x + \frac{3\pi}{8} \right) \) lies \( \frac{3\pi}{8} \) units to the left of the graph of \( y = \cos(x) \).

### Explanation:
The equation \( y = \cos \left( x + \frac{3\pi}{8} \right) \) represents a phase shift of the standard cosine function, \( y = \cos(x) \).

- A phase shift to the left happens when we have \( y = \cos(x + c) \), where \( c > 0 \).
- A phase shift to the right occurs when \( y = \cos(x - c) \).

In this case, \( x \) is replaced with \( x + \frac{3\pi}{8} \), which indeed shifts the graph to the left by \( \frac{3\pi}{8} \).

Therefore, the original statement is **true**.

### Answer:
- The correct answer is **True**.

Would you like more details on phase shifts or a deeper breakdown of this concept? Here are some follow-up questions to help reinforce your understanding:

1. What is the general rule for phase shifts in trigonometric functions?
2. How would the graph change if the equation was \( y = \cos \left( x - \frac{\pi}{4} \right) \)?
3. Can you describe how vertical shifts differ from phase shifts in trigonometric graphs?
4. What happens to the amplitude of the cosine function if we modify the equation to \( y = 2\cos(x + \frac{\pi}{4}) \)?
5. How would the graph of \( y = \cos(x) \) change if we applied a horizontal stretch instead of a phase shift?

**Tip:** Always remember that adding a constant inside the function argument causes horizontal shifts, while adding a constant outside affects vertical shifts.

Determine whether the following statement is true or false: The graph of y = cos(x + 3π/8) lies 3π/8 units to the left of the graph of y = cos(x).

This trigonometry problem examines the phase shift of the graph y = cos(x + 3π/8) compared to y = cos(x). Learn how phase shifts affect the position of cosine graphs using key trigonometric concepts and formulas. Suitable for high school students in Grades 10-12.

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