Problem 26 - 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. $$ \sin \theta<0, \tan \theta>0 $$

Step-by-Step Solution

Answer

The terminal side of \(\angle A O B\) lies in the third quadrant.

Step 1: Understanding Quadrants based on Sine

The sine function is associated with the y-coordinate of a point on the unit circle. If \(\sin \theta < 0\), it means the y-value is negative. Therefore, \(\theta\) lies either in the third or fourth quadrant.

Step 2: Understanding Quadrants based on Tangent

The tangent function, \(\tan \theta\), is positive when either both sine and cosine are positive (first quadrant) or both are negative (third quadrant). Given \(\tan \theta > 0\), the angle \(\theta\) could be in the first or third quadrant, provided that sine and cosine signs align correctly.

Step 3: Determining the Correct Quadrant

Since \(\sin \theta < 0\) (negative y-coordinate) and \(\tan \theta > 0\) (both sine and cosine have the same sign), the angle must be in the third quadrant where both sine is negative and tangent is positive.

Key Concepts

Understanding Sine Negative

The sine function is a fundamental part of trigonometry and plays a crucial role in determining the position of angles on the unit circle. The sine of an angle, denoted as \(\sin \theta\), represents the y-coordinate of the point where the angle's terminal side intersects the unit circle. This means that when \(\sin \theta < 0\), the y-coordinate is negative. In the context of the unit circle:

• \(\sin \theta > 0\) in the first and second quadrants, where the y-coordinates are above the x-axis.
• \(\sin \theta < 0\) in the third and fourth quadrants, where the y-coordinates are below the x-axis.

Therefore, if you know \(\sin \theta < 0\), you can immediately narrow down your options to the third or fourth quadrants. These are the only places on the unit circle where the sine value of an angle can be negative, aligning with a negative y-coordinate. To summarize, the key takeaway for a negative sine value is to only consider the third and fourth quadrants when tracking an angle on the unit circle.

Identifying Tangent Positive

The tangent function, represented as \(\tan \theta\), is found by dividing the sine of an angle by its cosine, or \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). The sign of the tangent function is determined by the relative signs of sine and cosine. For \(\tan \theta > 0\), both \(\sin \theta\) and \(\cos \theta\) must either be positive or negative.

• Tangent is positive in the first quadrant where both sine and cosine are positive.
• Tangent is also positive in the third quadrant where both sine and cosine are negative.

If you're given that \(\tan \theta > 0\), the angle \(\theta\) could be situated in either the first or third quadrant, based on the combination of signs of sine and cosine. This evaluation helps further narrow down the quadrant possibilities by requiring both sine and cosine to have matching signs for tangent to remain positive. By understanding these principles, you can easily determine the potential quadrants when confronted with a positive tangent value.

Exploring Unit Circle Quadrants

The unit circle is a powerful tool in trigonometry, offering a simple way to visualize and solve problems related to angles and trigonometric functions. It is a circle with a radius of one centered at the origin of a coordinate plane. This circle is divided into four quadrants, each dictating specific sign behaviors for sine, cosine, and tangent. Let's go over the standard quadrant breakdown:

• First Quadrant: Sine and cosine are both positive, so tangent is also positive.
• Second Quadrant: Sine is positive, cosine is negative, resulting in a negative tangent.
• Third Quadrant: Sine and cosine are both negative, therefore tangent is positive.
• Fourth Quadrant: Sine is negative, cosine is positive, and tangent is negative.

Knowing these quadrant properties helps resolve which specific quadrant an angle resides in given certain conditions. For instance, for an angle \(\theta\) where \(\sin \theta < 0\) and \(\tan \theta > 0\), you would position it in the third quadrant. These quadrant properties are essential for solving trigonometric problems and understanding the behavior of functions as they relate to angles on the unit circle. With practice, recognizing the distinct characteristics of each quadrant becomes second nature, making trigonometric problem-solving more intuitive.