Problem 60 - Algebra and Trigonometry

Question

Write the first expression in terms of the second if the terminal point determined by \(t\) is in the given quadrant. \(\csc t, \cot t ; \quad\) Quadrant III

Step-by-Step Solution

Answer

In Quadrant III, \(\csc t = \frac{-\cot t}{\sqrt{1 - \cot^2 t}}.\)

Step 1: Understanding the Given Values

We need to express \(\csc t\) in terms of \(\cot t\) given that the terminal point determined by \(t\) is in Quadrant III. In this quadrant, both sine and cosine are negative, thus making the cosecant (\(\csc t = \frac{1}{\sin t}\)) and cotangent (\(\cot t = \frac{\cos t}{\sin t}\)) negative as well.

Step 2: Expressing Sine in terms of Cotangent

Since \(\cot t = \frac{\cos t}{\sin t}\), we can solve for \(\sin t\): \[ \sin t = \frac{\cos t}{\cot t} \] Since \(\sin t\) is negative in Quadrant III, our expression becomes: \[ \sin t = -\frac{\cos t}{\cot t} \]

Step 3: Using the Pythagorean Identity

Recall the identity \(\sin^2 t + \cos^2 t = 1\). We want an expression for \(\cos t\), noting it will also be negative in Quadrant III. Substituting \(\cos t = -\sqrt{1 - \sin^2 t}\) gives us an in-term relation for \(\sin t\): \[ \sin t = -\frac{\sqrt{1 - \sin^2 t}}{\cot t} \]

Step 4: Simplifying the Expression

Now, to express \(\csc t\) in terms of \(\cot t\), substitute \(\sin t\) in \(\csc t = \frac{1}{\sin t}\): \[ \csc t = \frac{-\cot t}{\sqrt{1 - \cot^2 t}} \] This expresses \(\csc t\) in terms of \(\cot t\) with appropriate signs for Quadrant III.

Key Concepts

Cosecant function

The cosecant function, often abbreviated as "csc," is the reciprocal of the sine function. In mathematical terms, it is defined as \( \csc t = \frac{1}{\sin t} \). This means that for any angle \( t \), the cosecant is equal to one divided by the sine of that angle. This function is significant in trigonometry as it helps us find lengths in right triangles. Cosecant is considered undefined when \( \sin t \) is zero because division by zero is not possible in mathematics. Thus, wherever the sine value is zero, such as at integer multiples of \( \pi \), the cosecant also fails to exist. When evaluating \( \csc t \) specifically in Quadrant III, both sine and cosecant are negative. It's crucial to remember this sign change, as neglecting it can lead to confusion in calculations. Furthermore, due to its definition, \( \csc \) often takes larger values as \( \sin \) approaches zero. This relationship highlights the importance of understanding the behavior of the cosecant function beyond just its numerical computation.

Cotangent function

The cotangent function, or "cot" function, is closely related to both the tangent and sine functions. It is defined mathematically as \( \cot t = \frac{\cos t}{\sin t} \) or alternatively as the reciprocal of the tangent function \( \cot t = \frac{1}{\tan t} \). Understanding cotangent is pivotal in trigonometry, especially when dealing with angles.Since \( \cos t \) and \( \sin t \) are involved, cotangent captures the ratio of the adjacent side to the opposite side in a right triangle, providing an angle measurement in trigonometric calculations.In Quadrant III, both the sine and cosine functions have negative values. Therefore, the cotangent, as their ratio, becomes positive. Recognizing this sign change is essential for solving equations correctly. Moreover, like the tangent function, cotangent tends to be undefined where sine is zero, reiterating the reciprocal relationship between cotangent and tangent. This function thus provides a vital link in the circle of trigonometric identities.

Quadrant III

Understanding the behavior of trigonometric functions in Quadrant III is key for solving many trigonometry problems. The coordinate plane is divided into four quadrants and Quadrant III is where both x and y coordinates are negative. This means the sine and cosine functions, which relate to y and x respectively on the unit circle, are both negative in this quadrant.Consequently, trigonometric calculations in Quadrant III must consider these signs. As a result, the cosecant \( \csc t \) and sine \( \sin t \) are both negative, while the cotangent \( \cot t \) becomes positive due to its ratio of negative values, whereas tangent \( \tan t \), being the reciprocal of cotangent, is also positive.This quadrant uniquely influences the identity expressions of these functions. Knowledge of these properties aids in both theoretical analyses and practical computations in trigonometry, empowering students to tackle a range of problems with confidence and precision. Keeping these sign rules in mind helps avoid common mistakes during evaluations of trigonometric expressions.