Problem 22 - 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. $$ \sin \theta<0, \cos \theta>0 $$

Step-by-Step Solution

Answer

The terminal side of \( \angle AOB \) lies in Quadrant IV.

Step 1: Analyze the Sine Condition

The condition given is \( \sin \theta < 0 \). This indicates that \( \theta \) is in a quadrant where the sine function, which corresponds to the y-coordinate of the point on the unit circle, is negative. This occurs in Quadrants III and IV.

Step 2: Analyze the Cosine Condition

The condition given is \( \cos \theta > 0 \). This indicates that \( \theta \) is in a quadrant where the cosine function, which corresponds to the x-coordinate of the point on the unit circle, is positive. This occurs in Quadrants I and IV.

Step 3: Determine the Common Quadrant

We need to find the quadrant where both conditions are satisfied. The sine is negative in Quadrants III and IV, and cosine is positive in Quadrants I and IV. The common quadrant where both conditions \( \sin \theta < 0 \) and \( \cos \theta > 0 \) are true is Quadrant IV.

Key Concepts

Unit Circle

The Unit Circle is a fundamental concept in trigonometry that helps us understand angles and trigonometric functions easily. Imagine a circle with a radius of 1 unit, centered at the origin of a coordinate plane. This circle is special because every point on its circumference can be identified using coordinates that represent familiar trigonometric values. These values are based on the angle formed with the positive x-axis.
Each angle corresponds to a point on the unit circle, providing its cosine as the x-coordinate and its sine as the y-coordinate. As angles increase from 0 to 360 degrees (or 0 to 2π radians), their positions on the unit circle help define where they lie in terms of quadrants.
Understanding the quadrant where an angle lies is important because:

• Each quadrant has different positive and negative signs for sine and cosine.
• This helps determine the specific range of angles and the trigonometric behavior.

So, when given the signs of sine and cosine, the unit circle guides us to pinpoint which quadrant is relevant depending on which trigonometric function is positive or negative there.

Sine Function

The Sine Function is one of the primary functions in trigonometry and is crucial for solving problems involving angles. When we talk about the sine of an angle, we are referring to the y-coordinate of the angle’s endpoint on the unit circle. This coordinates range from -1 to 1 as it traces the circle.

• In Quadrant I, sine is positive because we have positive y-values.
• In Quadrant II, sine remains positive as the y-coordinate is still above the x-axis.
• In Quadrants III and IV, sine becomes negative, as y-coordinates fall below the x-axis.

The sine function has a periodic nature with a period of 2π, meaning it repeats every full rotation around the circle. This periodicity is useful in many applications involving cycles and oscillations, like sound waves or pendulum motions.

Cosine Function

The Cosine Function is another significant trigonometric function, closely related to the sine function. It represents the x-coordinate of a point on the unit circle at a certain angle. Like the sine, cosine values range from -1 to 1, describing how far left or right a point is from the center.

• Cosine is positive in Quadrants I and IV where x-values are positive.
• In Quadrants II and III, cosine is negative as the point lies to the left of the y-axis.

Cosine shares the same 2π period with sine, which signifies its repeating pattern after a full revolution around the circle. Recognizing the signs and behavior of the cosine function across different quadrants allows us to solve for angles and verify identities in various trigonometric problems.