Problem 90 - A Graphical Approach to Precalculus with Limits

Question

Identify the quadrant (or possible quadrants) of an angle \(\theta\) that satisfies the given conditions. $$\cos \theta<0, \sin \theta<0$$

Step-by-Step Solution

Answer

The angle \(\theta\) is in Quadrant III.

Step 1: Understanding the Problem

We need to find the quadrant(s) where both \(\cos \theta < 0\) and \(\sin \theta < 0\). Quadrants are defined on the unit circle in the coordinate plane, and each has different signs for sine and cosine.

Step 2: Analyzing Quadrant Conditions

In the coordinate plane, each quadrant has specific sign conditions for sine and cosine:- Quadrant I: \(\cos \theta > 0\), \(\sin \theta > 0\)- Quadrant II: \(\cos \theta < 0\), \(\sin \theta > 0\)- Quadrant III: \(\cos \theta < 0\), \(\sin \theta < 0\)- Quadrant IV: \(\cos \theta > 0\), \(\sin \theta < 0\)

Step 3: Identifying the Correct Quadrant

Given the conditions \(\cos \theta < 0\) and \(\sin \theta < 0\), we compare them to the conditions in each quadrant. Both conditions are met in Quadrant III where both \(\cos \theta < 0\) and \(\sin \theta < 0\).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. It's a circle with a radius of 1, centered at the origin of a coordinate plane. This circle is key for understanding how angles transform trig functions like sine and cosine. On the unit circle, any angle can be represented as a point \(x, y\) where x is \(\cos \theta\) and y is \(\sin \theta\).

• The entire circumference measures \(2\pi\) radians.
• Angles measured counterclockwise from the positive x-axis yield accurate sine and cosine values.

Knowing the unit circle helps determine where an angle lies based on sine and cosine values, making it easier to understand the properties of different quadrants.

Quadrants

The coordinate plane is divided into four quadrants for trigonometry, each helping us understand the sign of sine and cosine.

• Quadrant I: Both sine and cosine are positive (\(\sin \theta > 0\) and \(\cos \theta > 0\)).
• Quadrant II: Sine is positive, cosine is negative (\(\sin \theta > 0\) and \(\cos \theta < 0\)).
• Quadrant III: Both sine and cosine are negative (\(\sin \theta < 0\) and \(\cos \theta < 0\)).
• Quadrant IV: Sine is negative, cosine is positive (\(\sin \theta < 0\) and \(\cos \theta > 0\)).

These quadrants are crucial since the problem requires recognizing where both sine and cosine are negative, which occurs in Quadrant III.

Sine and Cosine

Sine and cosine are two primary trigonometric functions that represent the coordinates of points on the unit circle. The cosine of an angle represents its x-coordinate, while the sine represents its y-coordinate.

• Sine (\(\sin \theta\)): A trigonometric function giving the vertical position of the endpoint of an angle on the unit circle.
• Cosine (\(\cos \theta\)): A trigonometric function giving the horizontal position of the endpoint of an angle on the unit circle.

Their values are constrained between -1 and 1 as they are ratios of sides of a right triangle, which in the context of the unit circle, are bound by its radius.

Angle Identification

Identifying the correct quadrant of an angle can be a straightforward process using the unit circle and the properties of sine and cosine. In the exercise, \(\cos \theta < 0\) and \(\sin \theta < 0\) indicate which quadrant the angle belongs to by examining where these conditions are satisfied.

• Since both sine and cosine are negative in Quadrant III, the angle \(\theta\) lies there.
• Other quadrants have different sign conditions that necessarily exclude them for this angle.

Understanding how to match these trigonometric conditions with the corresponding quadrants on a unit circle is essential for angle identification in trigonometry.