Problem 47 - Precalculus Mathematics for Calculus

Question

Find the sign of the expression if the terminal point determined by \(t\) is in the given quadrant. \(\sin t \cos t, \quad\) Quadrant II

Step-by-Step Solution

Answer

The expression is negative in Quadrant II.

Step 1: Recall Quadrant Properties

In the Cartesian coordinate system, each of the four quadrants has distinct sign properties for the sine and cosine functions. In Quadrant II, the sine function is positive and the cosine function is negative.

Step 2: Identify Function Values' Signs

Since the terminal point determined by \(t\) is in Quadrant II, we know that \(\sin t > 0\) and \(\cos t < 0\).

Step 3: Determine the Sign of the Product

The expression we need to evaluate is \(\sin t \cos t\). The product of two numbers is positive if both numbers have the same sign and negative if the numbers have opposite signs. Since \(\sin t > 0\) and \(\cos t < 0\) in Quadrant II, the signs are opposite.

Step 4: Conclude Sign Based on Different Signs

Since \(\sin t\) and \(\cos t\) have opposite signs, their product \(\sin t \cos t\) will be negative.

Key Concepts

Quadrant Properties

The Cartesian coordinate system is divided into four quadrants. Each quadrant has unique sign characteristics for the trigonometric functions like sine (\( ext{sin}\)) and cosine (\( ext{cos}\)). Here's how the sign of these functions is determined by the quadrant:

• Quadrant I: Both sine and cosine are positive. This is where both x and y coordinates are positive.
• Quadrant II: Sine is positive, but cosine is negative, because the x-values are negative.
• Quadrant III: Both sine and cosine are negative since both the x and y coordinates are negative.
• Quadrant IV: Sine is negative, and cosine is positive for positive x-values.

Remembering these properties helps predict the behavior of trigonometric functions. For any angle that terminates in Quadrant II, like in our original exercise, it is critical to recall that sine will be positive and cosine will be negative.

Sine Function

The sine function is a fundamental trigonometric function that describes the y-coordinate of a point on the unit circle. In simpler terms, when you have an angle, sine tells you how high or low the tip of the angle is along the vertical axis.

In the context of the Cartesian coordinate system, the sine function has specific sign properties based on the quadrants:

• Positive in Quadrant I and II.
• Negative in Quadrant III and IV.

Understanding sine's behavior in different quadrants helps solve various trigonometric problems. In Quadrant II, as mentioned in the exercise, sine is positive because you are dealing with angles that bring you above the x-axis.

Cosine Function

The cosine function is another fundamental trigonometric function that represents the x-coordinate of a point on the unit circle. Essentially, it tells you how far right or left the tip of the angle is along the horizontal axis.

Just like the sine function, cosine has its own sign properties depending on which quadrant the angle falls into:

• Positive in Quadrant I and IV.
• Negative in Quadrant II and III.

In the second quadrant, where we find our angle from the original exercise, the cosine is negative. This is because in this region, angles correspond to negative x-values (left side of the y-axis).

Cartesian Coordinate System

The Cartesian coordinate system is a two-dimensional plane used for graphing and locating points through a grid of x (horizontal) and y (vertical) axes.

Here’s a quick overview of its main features:

• The system is divided into four quadrants based on the x and y axes.
• Quadrants are numbered counterclockwise starting from the top right: I, II, III, and IV.
• Each quadrant gives different sign properties to the trigonometric functions.

Knowing how to use the Cartesian coordinate system allows you to identify the sign of trigonometric functions depending on which quadrant the angle lies in. In context, Quadrant II is characterized by a positive y-axis and a negative x-axis.