The equation given is \( t - \sin(t) = 0 \).

To solve for \( t \), let's consider the equation:

\[
t = \sin(t)
\]

This equation implies that the value of \( t \) must be equal to its sine value. 

### Numerical Solution

This type of equation generally does not have an exact analytical solution, so we must solve it numerically. Let's analyze it graphically or use a numerical method like Newton's method to approximate the solutions.

### Graphical Analysis

Consider the graphs of \( y = t \) and \( y = \sin(t) \):

- \( y = t \) is a straight line passing through the origin with a slope of 1.
- \( y = \sin(t) \) is a periodic wave oscillating between \(-1\) and \(1\).

The points of intersection between these two graphs will give the solutions to the equation \( t - \sin(t) = 0 \).

**Solution:**

1. **\( t = 0 \)** is an obvious solution because \( \sin(0) = 0 \).
2. There are no other simple intersections where \( t \) and \( \sin(t) \) are equal, as \( t = \sin(t) \) can only intersect within the range of the sine function, which is between \(-1\) and \(1\).

Given the nature of the sine function and the line \( t = \sin(t) \), \( t = 0 \) is the only real solution.

### Final Answer:
\[ t = 0 \]

Would you like more details or have any questions?

**Relative Questions:**
1. How would you solve \( t - \cos(t) = 0 \)?
2. What is the general method to solve transcendental equations like this?
3. How can Newton's method be applied to solve \( t - \sin(t) = 0 \) numerically?
4. Can you explain why \( t = \sin(t) \) has limited intersections?
5. What are the properties of the sine function that affect the solutions?

**Tip:** For transcendental equations, numerical methods like Newton's or graphical approaches often provide the most practical solutions.

Explore how to solve the transcendental equation t - sin(t) = 0 using numerical and graphical methods. This problem involves understanding transcendental equations and applying the Intermediate Value Theorem. Suitable for advanced high school students and beyond.

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