To find $\tan \left( \frac{5\pi}{4} \right)$ without a calculator, you can follow these steps:

Step 1: Identify the angle's reference angle and quadrant

First, recognize that $\frac{5\pi}{4}$ is in radians. To better understand this angle, you can convert it into degrees:

$\frac{5\pi}{4} \times \frac{180^\circ}{\pi} = 225^\circ$

This is an angle in the third quadrant (since 225° is between 180° and 270°).

Step 2: Find the reference angle

The reference angle is the smallest angle between the given angle and the x-axis. For 225°, the reference angle is:

$225^\circ - 180^\circ = 45^\circ$

So, the reference angle is $45^\circ$ or $\frac{\pi}{4}$ radians.

Step 3: Determine the sign of the tangent

In the third quadrant, both sine and cosine are negative, but since $\tan \theta = \frac{\sin \theta}{\cos \theta}$, the negative signs cancel out, making the tangent positive.

Step 4: Use the known value for $\tan \left( \frac{\pi}{4} \right)$

We know that:

$\tan \left( \frac{\pi}{4} \right) = 1$

Therefore, since the reference angle is $\frac{\pi}{4}$ and the tangent is positive in the third quadrant, we conclude that:

$\tan \left( \frac{5\pi}{4} \right) = 1$

Final Answer:

$\tan \left( \frac{5\pi}{4} \right) = 1$

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Would you like further details or clarifications on any of the steps? Here are some related questions to explore:

1. What is the unit circle, and how does it help in finding trigonometric values?
2. How do the signs of trigonometric functions change in different quadrants?
3. What are the exact values of trigonometric functions for common angles like $\frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}$?
4. How do you convert radians to degrees and vice versa?
5. How can we find the tangent of other angles, such as $\frac{7\pi}{6}$?

Tip: To quickly determine the sign of trigonometric functions in each quadrant, remember the acronym "All Students Take Calculus"—where each letter represents whether sine, cosine, and tangent are positive in each of the four quadrants.