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Understanding the basic trigonometric functions sine, cosine, and tangent is key to analyzing angles and their relationships with right triangles.

The sine of an angle, often noted as \(sin\), is the ratio between the length of the opposite side of the angle and the hypotenuse in a right triangle. Similarly, cosine, \(cos\), is the ratio between the adjacent side and the hypotenuse. Lastly, tangent, \(tan\), is the ratio of the opposite side to the adjacent side.

These ratios are fundamental in determining an angle's characteristics without the need for measurement. In the example \(11 \pi / 4\), evaluating each function's value for this angle involves understanding their behavior in different quadrants of the unit circle.

Trigonometric functions are periodic, meaning they repeat their values in regular intervals. For instance, the sine and cosine functions have a period of \(2\pi\), while the tangent function has a period of \(\pi\).

To simplify complex angle measurements like \(11 \pi / 4\), we exploit this periodicity to find an equivalent angle within the fundamental period (0 to \(2\pi\)). This way, it is easier to evaluate trigonometric functions without altering the angle's terminal side position on the unit circle.

By normalizing \(11 \pi / 4\) to \(3\pi / 4\), we managed to get an angle within the primary range, allowing straightforward evaluation using known reference angles.

Radian measure is an alternative to degrees for quantifying angles, crucial in the realm of trigonometry. One radian is the angle made at the center of a circle by an arc whose length is equal to the circle's radius.

In our example, the angle is given in radians as \(11\pi / 4\). There are \(2\pi\) radians in a full circle (similar to 360 degrees), so converting between radians and degrees—and vice-versa—is a common exercise when evaluating trig functions.

Working with radians provides a natural connection between linear and angular motion, which is why they are often preferred in mathematics and the sciences. Being comfortable with radian measure allows you to easily navigate and normalize angles for analysis, just as done in the normalized angle step of the solution provided.

Trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. They are invaluable tools for simplifying and solving trigonometric expressions. Some of the basic identities include the Pythagorean identities, angle sum and difference identities, and double angle formulas.

When evaluating trigonometric functions at an angle like \(3\pi / 4\), one might use the identity \(\sin(\theta) = \cos(\pi/2 - \theta)\) to find sine values by knowing the cosine of the complementary angle. These identities are vital for understanding and computing trigonometric functions efficiently, as demonstrated in converting the functions of \(11 \pi / 4\) to the simple functions of \(3 \pi / 4\).