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.