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The Pythagorean theorem is a fundamental principle in geometry that describes the relationship between the sides of a right triangle. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This relationship is typically written as \(a^2 + b^2 = c^2\), where \(a\) and \(b\) are the lengths of the legs, and \(c\) is the length of the hypotenuse.

In the given exercise, the tangent of angle \(\theta\) is used to find the lengths of the opposite and adjacent sides of the triangle. Once we know these two sides, we use the Pythagorean theorem to calculate the length of the hypotenuse, which is essential in finding the sine and cosine values.

Understanding the Pythagorean theorem is not only critical for solving many problems in trigonometry but also for appreciating the properties of space and shapes. It's often one of the first steps in finding the values of trigonometric functions when you know the lengths of any two sides of a right triangle.

The unit circle is a powerful tool in trigonometry that helps visualize and understand the trigonometric functions. It is a circle with a radius of one unit, centered at the origin of a coordinate plane. The unit circle allows for the definition of the trigonometric functions cosine, sine, and tangent based on the coordinates of points along its circumference.

Each point on the unit circle corresponds to an angle measured from the positive x-axis, going counter-clockwise. The coordinates of these points \( (x, y) \) provide the values of \( \cos(\theta) \) and \( \sin(\theta) \) for the angle \(\theta\). Specifically, \(\cos(\theta) = x\) and \(\sin(\theta) = y\). The quadrant in which the terminal side of the angle lies will determine the sign of those trigonometric values.

In our exercise, knowing that \(\cos(\theta) < 0\) implies that angle \(\theta\) must lie in either the second or third quadrant, where the x-coordinates (and thus the cosine values) are negative. The unit circle is not just a theoretical construct but is used practically to solve problems and understand trigonometric function properties and their periodic nature.

Trigonometry isn't only about the primary functions like sine, cosine, and tangent; understanding their reciprocal functions is equally important for a comprehensive grasp of the subject. The reciprocal trigonometric functions are secant (\(\sec\)), cosecant (\(\csc\)), and cotangent (\(\cot\)), which are reciprocals of cosine, sine, and tangent, respectively.

Specifically, \(\sec(\theta) = \frac{1}{\cos(\theta)}\), \(\csc(\theta) = \frac{1}{\sin(\theta)}\), and \(\cot(\theta) = \frac{1}{\tan(\theta)}\). These functions are useful when the primary trigonometric functions' values become inconvenient or impractical to use. They are also critical in various areas of calculus and applied mathematics.

In the provided solution, after finding the values of sine and cosine, the reciprocal functions' values are determined. For example, since \(\cos(\theta)\) was found to be negative, \(\sec(\theta)\) will also be negative, as they are reciprocal. Understanding these relationships can unlock further aspects of trigonometry in both theoretical and real-world applications.