Problem 52 - Precalculus Mathematics for Calculus

Question

From the information given, find the quadrant in which the terminal point determined by \(t\) lies. \(\tan t > 0\) and \(\sin t < 0\)

Step-by-Step Solution

Answer

The terminal point determined by \( t \) lies in the third quadrant.

Step 1: Identify the Quadrants for Tangent

The tangent function, \( \tan t \), is positive in the first and third quadrants. Therefore, the possible quadrants for \( \tan t > 0 \) are the first or third quadrant.

Step 2: Identify the Quadrants for Sine

The sine function, \( \sin t \), is negative in the third and fourth quadrants. Therefore, the possible quadrants for \( \sin t < 0 \) are the third or fourth quadrant.

Step 3: Intersection of Conditions

Since both conditions must be satisfied simultaneously, we look for the quadrant common to both scenarios. The only quadrant where both \( \tan t > 0 \) and \( \sin t < 0 \) is true is the third quadrant.

Key Concepts

Quadrants

The concept of quadrants is essential in understanding trigonometric functions. A quadrant refers to one of the four sections into which the coordinate plane is divided. Each quadrant corresponds to a specific range of angles:

• First Quadrant: angles between 0° and 90°
• Second Quadrant: angles between 90° and 180°
• Third Quadrant: angles between 180° and 270°
• Fourth Quadrant: angles between 270° and 360°

These quadrants are numbered counter-clockwise starting from the positive x-axis. Understanding which trigonometric function is positive in each quadrant helps solve exercises involving angles and their terminal sides located in different quadrants.

Tangent Function

The tangent function, represented as \( an \), is one of the primary trigonometric functions. Its value is defined as the ratio of the sine of an angle to its cosine: \( an heta = \frac{\sin \theta}{\cos \theta} \).

The sign of the tangent function depends on the signs of the sine and cosine functions:

• First Quadrant: both sine and cosine are positive, making tangent positive.
• Second Quadrant: sine is positive and cosine is negative, making tangent negative.
• Third Quadrant: both sine and cosine are negative, making tangent positive.
• Fourth Quadrant: sine is negative and cosine is positive, making tangent negative.

Noticing these patterns allows you to determine the sign of tangent easily referred to its given condition, such as in the exercise where \( an t > 0 \).

Sine Function

The sine function (\( \sin \)) measures the vertical component of an angle on the unit circle. Its values fluctuate between -1 and 1 and are given by the y-coordinate of the terminal point on the unit circle.

Let's explore which quadrants have negative sine values:

• First Quadrant: sine is positive.
• Second Quadrant: sine is positive.
• Third Quadrant: sine is negative.
• Fourth Quadrant: sine is negative.

As seen, sine values are negative in the third and fourth quadrants. Combined with information about tangent, this helps locate the specific quadrant a particular angle terminates in, which, for \( an t > 0 \) and \( \sin t < 0 \), is the third quadrant according to the exercise.