91Ó°ÊÓ

The tangent addition formula is a powerful tool in trigonometry. It's particularly helpful for simplifying expressions that involve the tangent of sums or differences of angles. The formula is given by:

• \( \tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)} \).

When you see an equation involving tangents added together, this formula might be a key step to solving it. In our particular problem, we can identify the double angle scenario, since we have terms involving \( \tan(2t) \) and \( \tan(t) \).

By setting \( a = t \) and \( b = t \) (thus \( a + b = 2t \)), the formula helps us express \( \tan(2t) \) as a combination of \( \tan(t) \):

• \( \tan(2t) = \frac{2\tan(t)}{1 - \tan^2(t)} \).

This transformation is essential for reducing the problem into a simpler form that can be more easily solved.

Solving trigonometric equations involves finding the angle values that satisfy a given trigonometric identity. In our case, we started with an equation:

• \( \tan(2t) + \tan(t) = 1 - \tan(2t)\tan(t) \).

Initially, we applied the tangent addition formula to simplify the left-hand side of the equation. Once simplified, this leads us to new forms of the original equation, focusing primarily on single terms like \( \tan(3t) = 0 \).

To solve \( \tan(3t) = 0 \), we recognize that \( \tan(x) = 0 \) whenever \( x = n\pi \). So, this equation becomes:

• \( 3t = n\pi \), for integer values of \( n \).

Solving for \( t \), we have:

• \( t = \frac{n\pi}{3} \).

From this general solution, several specific values of \( t \) can be calculated within the required interval.

Interval notation is a shorthand used to describe the set of all numbers within a certain interval on the number line. This directive is often used in solving equations to limit the solutions to a specific section of the graph.

In this problem, we focus on the interval \([0, \pi)\), which includes all numbers from 0 up to but not including \( \pi \). When solving \( \tan(3t) = 0 \), we derive a general solution form, \( t = \frac{n\pi}{3} \).

• By substituting integers for \( n \), we find specific candidate solutions for \( t \).
• Only values of \( t \) that fall within \([0, \pi)\) are valid.

For example:

• \( t = 0, \frac{\pi}{3}, \) and \( \frac{2\pi}{3} \) are all solutions found within the interval.

This ensures our solutions are applicable to the constraints of the original problem.