Problem 25 - Algebra 2 and Trigonometry

Question

In \(21-26,\) if \(\theta\) is the measure of \(\angle A O B,\) an angle in standard position, name the quadrant in which the terminal side of \(\angle A O B\) lies. $$ \tan \theta<0, \cos \theta<0 $$

Step-by-Step Solution

Answer

The terminal side of \( \angle AOB \) lies in the second quadrant.

Step 1: Analyze the conditions

The problem states that \( \tan \theta < 0 \) and \( \cos \theta < 0 \). These conditions will help us identify the quadrant. The tangent function is negative in the second and fourth quadrants because tangent is positive in the first and third quadrants. The cosine function is negative in the second and third quadrants because cosine is positive in the first and fourth quadrants.

Step 2: Determine intersection of conditions

From Step 1, we determined that tangent is negative in the second and fourth quadrants, and cosine is negative in the second and third quadrants. The quadrant that satisfies both \( \tan \theta < 0 \) and \( \cos \theta < 0 \) is the second quadrant because it is the only quadrant where both conditions are met.

Key Concepts

Understanding Angle Quadrants

In trigonometry, a quadrant is a section of the Cartesian coordinate plane. The coordinate plane is divided into four quadrants by the x-axis and y-axis. Each quadrant represents a unique combination of signs for the sine, cosine, and tangent functions.

• Quadrant I: Both sine and cosine are positive, making tangent positive.
• Quadrant II: Sine is positive while cosine is negative, making tangent negative.
• Quadrant III: Both sine and cosine are negative, making tangent positive.
• Quadrant IV: Sine is negative while cosine is positive, making tangent negative.

The angle \(\theta\) is positioned based on where its terminal side lands, thus determining its quadrant. The four quadrants are labeled in a counter-clockwise manner starting from the positive x-axis.

Exploring the Tangent Function

The tangent of an angle \(\theta\) in a right triangle is defined by the ratio of the opposite side to the adjacent side. In terms of the unit circle, it is expressed as \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).
Understanding the signs of trigonometric functions helps us identify when tangent is positive or negative. In quadrants where sine and cosine have opposite signs, the tangent will be negative. Specifically:

• Quadrant II: Positive sine and negative cosine result in negative tangent.
• Quadrant IV: Negative sine and positive cosine also result in negative tangent.

Recognizing these properties is crucial when determining the position of angles on the coordinate plane.

Insights on the Cosine Function

Cosine, another fundamental trigonometric function, is represented as the ratio of the adjacent side to the hypotenuse in right triangle trigonometry. On the unit circle, it is the x-coordinate of a point where the terminal side of the angle intersects.
Key characteristics of the cosine function include:

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

Understanding where cosine is negative helps us identify that angles whose terminal sides lie in Quadrants II or III will have a negative cosine.
When combined with tangent properties, this insight allows the second quadrant to satisfy both \(\tan \theta < 0\) and \(\cos \theta < 0\).

Angles in Standard Position

An angle is said to be in "standard position" when its vertex is at the origin of the coordinate plane and its initial side lies along the positive x-axis.
The terminal side of the angle then rotates counterclockwise or clockwise to determine its measure. Thus:

• Counterclockwise rotation results in positive angles.
• Clockwise rotation results in negative angles.

Understanding standard position is essential for visualizing and solving problems related to trigonometric angles. Here, observing the quadrant into which the terminal side falls directly aids in identifying the characteristics of sine, cosine, and tangent for that angle.
This visualization is a central aspect of solving problems involving angles and their respective trigonometric function evaluations.