Problem 76 - Algebra 2

Question

In which quadrants are the solutions to \(\tan \theta+1=0 ?\) F. Quadrants I and II H. Quadrants II and IV G. Quadrants II and III J. Quadrants III and IV

Step-by-Step Solution

Answer

The solutions to the equation \(\tan \theta+1=0\) are in Quadrants II and IV (H).

Step 1: Equation Analysis

The given equation is \(\tan \theta+1=0 \). We need to solve for \(\theta\). Rearranging, we get \(\tan \theta = -1\). The goal is to find the quadrants where tangent function would yield -1.

Step 2: Recognize Tangent range

Recall that \(\tan \theta\) is positive in quadrants I and III and negative in quadrants II and IV. Given \(\tan \theta = -1\), this implies that solutions \(\theta\) would be in the quadrants where tangent is negative, hence quadrants II and IV.

Step 3: Solution Confirmation

To confirm, remember the unit circle. At angles of \(\frac{3\pi}{4}\) and \(\frac{7\pi}{4}\) (which are in the second and fourth quadrants respectively), tangent of the angles equals -1. Therefore, this confirms that the solutions lie in quadrants II and IV.

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. Imagine a circle with a radius of 1, centered at the origin of a coordinate plane. The unit circle allows us to define trigonometric functions for all angles.
When an angle, \( \theta \), is formed by a line from the origin intersecting the circle, its coordinates (\(x, y\)) correspond to (\(\cos \theta, \sin \theta\)). All key angles are points on this circle, like \(\frac{\pi}{4}\), \(\frac{3\pi}{4}\), \(\frac{5\pi}{4}\), and so on. These points make it easier to visualize and calculate trigonometric values.

• The unit circle is essential because it provides a model to understand how trigonometric functions behave.
• All angles around the circle correspond to specific points with \(x, y\) values linking directly to cosine and sine.
• This representation helps us understand periodicity and symmetry in trigonometric functions.

By using the unit circle, we can better grasp concepts like angles, periodicity, and function relationships.

Tangent Function

The tangent function, \( \tan \theta \), is crucial in trigonometry. It's defined as the ratio of the sine and cosine: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). This function is unique because it repeats every \(\pi\) radians.
The tangent line becomes undefined when \( \cos \theta = 0 \), such as at \(\pi/2\) and \(3\pi/2\), creating vertical asymptotes. For some angles like \(\frac{\pi}{4}\), \( \tan \theta = 1 \), and for \(\frac{3\pi}{4}\), \( \tan \theta = -1 \). These values inform how tangible results align with specific angle positions on the unit circle.

• Tangent is positive when both \( \sin \theta \) and \( \cos \theta \) share the same sign, such as in quadrants I and III.
• Tangent is negative when the signs are opposite, as seen in quadrants II and IV.

Understanding this helps in solving equations involving tangent. It was crucial in the problem where \( \tan \theta = -1 \) led us to identify quadrants II and IV.

Quadrants

The concept of quadrants is vital in analyzing trigonometric functions. The coordinate plane is divided into four quadrants, each having distinct characteristics. Quadrants help determine the signs and behavior of trigonometric functions.

• Quadrant I: Both \(x\) and \(y\) are positive. Trigonometric functions \( \sin, \cos, \tan \) are positive.
• Quadrant II: \(x\) is negative, \(y\) is positive. \(\sin\) is positive, and \(\cos, \tan\) are negative.
• Quadrant III: Both \(x\) and \(y\) are negative. \(\tan\) is positive, whereas \(\sin, \cos\) are negative.
• Quadrant IV: \(x\) is positive, \(y\) is negative. \(\cos\) is positive and \(\sin, \tan\) are negative.

In the given exercise, recognizing where \(\tan \theta = -1\) allowed us to pinpoint quadrants II and IV, embodying negative tangent values.
Familiarity with quadrant properties simplifies solving trigonometric equations across different angles.