Problem 51 - Algebra and Trigonometry

Question

From the information given, find the quadrant in which the terminal point determined by \(t\) lies. $$ \csc t>0 \text { and } \sec t<0 $$

Step-by-Step Solution

Answer

The terminal point lies in the second quadrant.

Step 1: Understand Trigonometric Signs

Recall that in trigonometry, \(\csc \, t = \frac{1}{\sin \, t}\), so \(\csc \, t > 0\) means \(\sin \, t > 0\). Similarly, \(\sec \, t = \frac{1}{\cos \, t}\), so \(\sec \, t < 0\) implies \(\cos \, t < 0\).

Step 2: Determine Possible Quadrants for Sine

Since \(\sin \, t > 0\), the sine function is positive in the first and second quadrants.

Step 3: Determine Possible Quadrants for Cosine

Since \(\cos \, t < 0\), the cosine function is negative in the second and third quadrants.

Step 4: Identify the Common Quadrant

We need a quadrant where both conditions are satisfied: \(\sin \, t > 0\) and \(\cos \, t < 0\). This is true only in the second quadrant.

Key Concepts

Understanding the Cosecant Function

The cosecant function, often denoted as \( \csc t \), is the reciprocal of the sine function. This means that \( \csc t = \frac{1}{\sin t} \). Because it is the reciprocal, when sine is positive, cosecant is also positive, and vice-versa.

It's important to note that the sine function is positive in two quadrants: the first and the second. Hence, \( \csc t > 0 \) means you're either in the first or second quadrant. Memorizing this can be helpful when trying to figure out where an angle’s terminal side might lie.

Understanding where the sine, and thus the cosecant, function is positive or negative, can aid in identifying quadrants for various trigonometric exercises. Practicing with different angles and evaluating their sine values will make you more comfortable with the cosecant function.

The Role of the Secant Function

The secant function, symbolized as \( \sec t \), is the reciprocal of the cosine function. It's defined as \( \sec t = \frac{1}{\cos t} \). Since secant is the reciprocal, when cosine is negative, secant is also negative, and when cosine is positive, so is secant.

Cosine is positive in the first and fourth quadrants, but negative in the second and third. This means \( \sec t < 0 \) indicates that we're dealing with either the second or third quadrant. This relationship is crucial when analyzing angles and their respective positions in a coordinate plane.

Knowing where the cosine and consequently the secant function are positive or negative helps in solving many trigonometric problems, especially those involving the identification of angle quadrants.

Decoding Trigonometric Identities

Trigonometric identities are relationships between the trigonometric functions. They are vital tools in simplifying and solving trigonometry problems. They can help you connect various trigonometric functions with each other, like knowing \( \csc t = \frac{1}{\sin t} \) and \( \sec t = \frac{1}{\cos t} \).

Understanding these identities enables you to determine the signs of functions in different quadrants, which is valuable for exercises like recognizing which quadrant an angle falls into. Moreover, identities like Pythagorean, angle sum and difference, and reciprocal identities, all serve as building blocks for more complex trigonometric analysis.

Therefore, mastering trigonometric identities not only aids in computing values but also in grasping the bigger picture of trigonometric relationships. Practicing these identities and applying them in various contexts will strengthen your understanding of trigonometry.