Problem 51 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Determine Possible Quadrants for Sine

The sine of an angle is positive in the first and second quadrants. This is because the sine function, which represents the y-coordinate on the unit circle, is positive above the x-axis.

Step 2: Determine Possible Quadrants for Cosine

The cosine of an angle is negative in the second and third quadrants. This occurs because the cosine represents the x-coordinate on the unit circle, which is negative on the left side of the y-axis.

Step 3: Find Common Quadrant

From previous findings, - \( \sin t > 0 \) is satisfied in Quadrants I and II.- \( \cos t < 0 \) is satisfied in Quadrants II and III.The only quadrant that satisfies both conditions is Quadrant II.

Key Concepts

Sine Function

The sine function is an essential part of trigonometry. It helps us find specific values related to angles, particularly in the unit circle. When you look at the unit circle, the sine of an angle, denoted as \( \sin t \), is the y-coordinate of a point on the circle, where \( t \) is the angle measured from the positive x-axis.

• Positive Sine Region: The sine function is positive in the first and second quadrants of the coordinate plane. This is because these quadrants lie above the x-axis, where y-coordinates are positive.
• Negative Sine Region: Conversely, sine is negative in the third and fourth quadrants, found below the x-axis.

To determine if \( \sin t \) is positive or negative, you need to identify the coordinate location of the angle’s terminal point within these quadrants. Remember, the unit circle is perfectly symmetrical around the origin, so changes in one coordinate (x or y) affect the signs of the trigonometric functions.

Cosine Function

The cosine function complements the sine function in analyzing angles and points on the unit circle. Cosine, represented as \( \cos t \), focuses on the x-coordinate of a point, giving insights into which part of the coordinate plane the angle’s terminal side lies.

• Positive Cosine Region: In the first and fourth quadrants, cosine is positive because the x-coordinate is positive as we move to the right of the y-axis.
• Negative Cosine Region: Cosine is negative in the second and third quadrants, where x-coordinates are found to the left of the y-axis.

When determining the cosine value for an angle, observing which quadrant the angle’s terminal side lands in helps establish whether \( \cos t \) is positive or negative. The sign of cosine helps in pinpointing the correct quadrant along with other trigonometric considerations.

Unit Circle

The unit circle is a powerful tool in trigonometry that provides a visual representation of angles and their corresponding trigonometric function values.

• Definition: A unit circle is a circle with a radius of one, centered at the origin (0, 0) of the coordinate plane. Angles, usually measured in radians, are mapped onto this circle starting from the positive x-axis.
• Coordinates On the Circle: Every point on the unit circle corresponds to a unique angle \( t \). The coordinates (x, y) of these points are the values of \( \cos t \) and \( \sin t \) respectively.

By understanding the unit circle, you can easily determine the characteristics of sine and cosine functions. The unit circle helps you visualize which quadrant an angle falls into based on the signs of its sine and cosine values. It's a fundamental concept that supports learning in both basic and advanced trigonometry, ensuring that you grasp the relationships between angles, their terminal points, and trigonometric functions.