91Ó°ÊÓ

The cosecant function, denoted as \(\text{csc } \theta\), is one of the six primary trigonometric functions. It is the reciprocal of the sine function. This means: \(\text{csc } \theta = \frac{1}{\text{sin } \theta}\). If you know the value of sine, you can easily find the cosecant by taking its reciprocal. In our problem, we are given \(\text{csc } \theta = -\frac{5}{2}\). This automatically tells us that \(\text{sin } \theta = -\frac{2}{5}\).

A fundamental theorem in trigonometry is the Pythagorean identity. It is represented as: \(\text{sin}^2 \theta + \text{cos}^2 \theta = 1\). This identity is incredibly useful for finding one trigonometric function when another is known. In our solution, we use this identity to find \(\text{cos } \theta\) given \(\text{sin } \theta = -\frac{2}{5}\). Substitute the value into the identity: \[ \begin{align*} \text{sin}^2 \theta + \text{cos}^2 \theta &= 1 \ \ \text{sin}^2 \theta + \text{cos}^2 \theta &= 1 \ \ \frac{4}{25} + \text{cos}^2 \theta &= 1 \ \ \text{cos}^2 \theta &= 1 - \frac{4}{25} \ \ \text{cos}^2 \theta &= \frac{21}{25} \ \ \text{cos } \theta &= -\frac{\text{\text{sqrt}}{21}}{5} \end{align*} \]. Note that we take the negative root because \(\theta\) is in Quadrant III where cosine is negative.

Trigonometric functions have reciprocals that often simplify calculations. The cosecant, secant, and cotangent functions are the reciprocals of sine, cosine, and tangent respectively.
To summarize:

• Cosecant (csc) - \(\text{csc } \theta = \frac{1}{\text{sin } \theta}\)
• Secant (sec) - \(\text{sec } \theta = \frac{1}{\text{cos } \theta}\)
• Cotangent (cot) - \(\text{cot } \theta = \frac{1}{\text{tan } \theta}\)

Given \(\text{cos } \theta = -\frac{\text{sqrt}}{21}}{5}\), the secant function is found as follows: \( \text{sec } \theta = -\frac{5 \text{\text{sqrt}}}{21}}{21}\). Similarly, the cotangent function is derived by: \( \text{cot } \theta = \frac{\text{sqrt}}{21}}{2}\). These reciprocal relationships help verify and cross-check other function values.

In trigonometry, knowing which quadrant an angle lies in can tell us a lot about the sign of each trigonometric function. The coordinate plane is divided into four quadrants:

• Quadrant I - All trigonometric functions are positive.
• Quadrant II - Sine is positive, but cosine and tangent are negative.
• Quadrant III - Both sine and cosine are negative, making tangent positive.
• Quadrant IV - Cosine is positive, but sine and tangent are negative.

In our problem, \( \theta \) falls in Quadrant III. Therefore, we know that both \(\text{sin } \theta\) and \(\text{cos } \theta\) will be negative. This is why \(\text{cos } \theta = -\frac{\text{\text{sqrt}}}{21}{5}\) was chosen with a negative sign.