Problem 21 - Precalculus Essentials

Question

In Exercises \(17-22\), let \(\theta\) be an angle in standard position. Name the quadrant in which \(\theta\) lies $$\tan \theta<0, \cos \theta<0$$

Step-by-Step Solution

Answer

The quadrant in which \(\theta\) lies is the 2nd quadrant.

Step 1: Identifying quadrants where cos is negative

Using the knowledge of trigonometric functions, identify the quadrants where cos \(\theta\) is negative. Cosine function is negative in the 2nd and 3rd quadrants.

Step 2: Identifying quadrants where tan is negative

Next, identify the quadrants where tan \(\theta\) is negative. The tangent function is negative in the 2nd and 4th quadrants.

Step 3: Find the common quadrants

The quadrant where both the constraints (cos \(\theta\) < 0 and tan \(\theta\) < 0) are valid would be the solution. From steps 1 and 2, it's clear that the 2nd quadrant is the common quadrant where cos \(\theta\) and tan \(\theta\) are both negative.

Key Concepts

Angle in Standard Position

In trigonometry, an angle is considered to be in standard position when its vertex is at the origin of the coordinate plane and its initial side is on the positive x-axis. The angle is measured from the initial side to the terminal side, moving counterclockwise. If the angle is measured clockwise, it is considered negative.

Knowing the concept of standard position is essential because it helps determine the sign of trigonometric functions depending on the angle's location in the Cartesian plane's quadrants. Each quadrant corresponds to a range of angles:

• 1st Quadrant: 0 to 90 degrees or 0 to \( \frac{\pi}{2} \) radians
• 2nd Quadrant: 90 to 180 degrees or \( \frac{\pi}{2} \) to \( \pi \) radians
• 3rd Quadrant: 180 to 270 degrees or \( \pi \) to \( \frac{3\pi}{2} \) radians
• 4th Quadrant: 270 to 360 degrees or \( \frac{3\pi}{2} \) to \( 2\pi \) radians

Understanding these quadrants is crucial for determining in which quadrant certain trigonometric functions will be positive or negative.

Tangent Function Negative

The tangent function, denoted as \( \tan \theta \), is defined as the ratio of the sine of the angle to the cosine of the angle, expressed as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). The sign of \( \tan \theta \) depends on the signs of sine and cosine for the given angle in a specific quadrant.

The tangent function is negative when its numerator and denominator have different signs. This occurs in the:

• 2nd Quadrant, where sine is positive, and cosine is negative.
• 4th Quadrant, where sine is negative, and cosine is positive.

Knowing in which quadrants tangent is negative helps determine the angle's quadrant when given cosine and tangent's signs.

Cosine Function Negative

The cosine function, represented by \( \cos \theta \), identifies the horizontal coordinate of a point on the unit circle that an angle \( \theta \) creates. Depending on the quadrant, \( \cos \theta \) can be positive or negative.

Specifically, the cosine function is negative in these quadrants:

• 2nd Quadrant, where angles are between \( \frac{\pi}{2} \) and \( \pi \) radians.
• 3rd Quadrant, where angles are between \( \pi \) and \( \frac{3\pi}{2} \) radians.

Recognizing where the cosine function becomes negative is crucial in solving trigonometric equations and helps in figuring out the possible positions of an angle, especially when combined with other trigonometric properties.