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Cosine is one of the primary trigonometric functions, often abbreviated as \( \cos \). It relates the adjacent side and hypotenuse of a right-angled triangle. In a right triangle, the cosine of angle \( \theta \) is calculated by

• \( \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \)

In the context of this exercise, given that \( \sec \theta = \frac{13}{5} \), we know that

• \( \sec \theta = \frac{1}{\cos \theta} \)
• \( \cos \theta = \frac{1}{\sec \theta} \)

So by substituting the known value \( \sec \theta = \frac{13}{5} \), we find \( \cos \theta = \frac{5}{13} \). This involves reversing the secant function to find the cosine. Cosine is crucial for understanding how different angles relate to one another within a right triangle.

Sine, represented as \( \sin \), is another foundational trigonometric function. It deals with the relationship between the opposite side and the hypotenuse of a right-angle triangle. The formula to find sine is

• \( \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \)

In the given problem, we used the Pythagorean identity, which states

• \( \cos^2 \theta + \sin^2 \theta = 1 \)

By substituting \( \cos \theta = \frac{5}{13} \) into this identity, we solve for \( \sin \theta \). Solving this step-by-step leads to finding \( \sin \theta = \pm \frac{12}{13} \). Since the sign of sine should match that of secant (and cotangent), the positive value \( \frac{12}{13} \) is used. Sine helps determine the opposite side length when the hypotenuse is known.

Cotangent, symbolized as \( \cot \), is a trigonometric function that is less commonly discussed than sine and cosine. However, it plays a significant role in understanding angle relationships. Cotangent is the reciprocal of tangent and is expressed as

• \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)

From the exercise, we calculated \( \cot \theta \) by using the previously found values of \( \cos \theta = \frac{5}{13} \) and \( \sin \theta = \frac{12}{13} \). Substituting these values into the cotangent formula yields \( \cot \theta = \frac{5}{12} \). This provides us with a comprehensive picture of how cotangent fits into the triangle's side lengths. Cotangent is particularly helpful in mathematical scenarios involving complementary angles and periodic functions.