Problem 24 - Calculus Single Variable

Question

State which of the six trigonometric functions are positive when evaluated at \(\theta\) in the indicated interval. \(\theta \in(3 \pi / 2,2 \pi)\)

Step-by-Step Solution

Answer

Cosine and secant are positive in the given interval.

Step 1: Understanding the Unit Circle

The angle \(\theta\) is situated in the fourth quadrant of the unit circle because it is in the interval \((\frac{3\pi}{2}, 2\pi)\). The fourth quadrant corresponds to angles ranging from \(270^\circ\) to \(360^\circ\) or \((\frac{3\pi}{2}, 2\pi)\).

Step 2: Positive and Negative Trigonometric Functions in Each Quadrant

In the fourth quadrant, cosine and secant are positive, while sine, cosecant, tangent, and cotangent are negative. This is because in the fourth quadrant, the x-coordinate (cosine) is positive while the y-coordinate (sine) is negative on the unit circle.

Step 3: Identifying the Positive Functions

Based on the analysis of the fourth quadrant, \(\cos(\theta)\) and \(\sec(\theta)\) are the functions that are positive. The functions \(\sin(\theta)\), \(\csc(\theta)\), \(\tan(\theta)\), and \(\cot(\theta)\) are negative in this quadrant.

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. It's a circle with radius 1, centered at the origin of a coordinate plane. Understanding the unit circle is crucial because it provides a geometric way to define trigonometric functions.

• The x-coordinate of a point on the unit circle represents the cosine of the angle.
• The y-coordinate represents the sine of the angle.

For any angle \(\theta\) in standard position (angle measured from the positive x-axis), the corresponding point on the unit circle is \((\cos(\theta), \sin(\theta))\).
This makes it easier to visualize and calculate trigonometric values by simply focusing on the coordinates of the points formed by the circle.

Quadrants in Trigonometry

The unit circle is divided into four quadrants. Each quadrant corresponds to a range of angle measures, and the sign (+/-) of trigonometric functions changes as you move between quadrants.

• **First Quadrant**: Angles from \(0\) to \(\pi/2\) (or \(0^\circ\) to \(90^\circ\)). Here both sine and cosine are positive.
• **Second Quadrant**: Angles from \(\pi/2\) to \(\pi\) (or \(90^\circ\) to \(180^\circ\)). Sine is positive, cosine is negative.
• **Third Quadrant**: Angles from \(\pi\) to \(3\pi/2\) (or \(180^\circ\) to \(270^\circ\)). Both sine and cosine are negative.
• **Fourth Quadrant**: Angles from \(3\pi/2\) to \(2\pi\) (or \(270^\circ\) to \(360^\circ\)). Cosine is positive, sine is negative.

Understanding which quadrant an angle is in helps determine the signs of the trigonometric functions.

Positive and Negative Trigonometric Functions

In trigonometry, the sign of a function is crucial as it indicates the direction of the angle relative to the x-axis and y-axis. In the context of the unit circle, whether a trigonometric function is positive or negative in a certain quadrant depends on the coordinates:

• **Positive Functions**: If within a given quadrant, the value of x or y is positive, this indicates that the corresponding trigonometric functions (cosine for x, sine for y) are positive.
• **Negative Functions**: Conversely, if either coordinate is negative, the corresponding trigonometric function will be negative.

For example, in the fourth quadrant,

• Cosine (\(\cos(\theta)\)) and secant (\(\sec(\theta)\)) are positive because the x-coordinate is positive.
• Sine (\(\sin(\theta)\)) and cosecant (\(\csc(\theta)\)), tangent (\(\tan(\theta)\)) and cotangent (\(\cot(\theta)\)) are negative, because the y-coordinate is negative.

This understanding simplifies the process of evaluating trigonometric functions based on the position of the angle on the unit circle.