Problem 36 - Algebra and Trigonometry

Question

Find the quadrant in which \(\theta\) lies from the information given. $$ \tan \theta<0 \quad \text { and } \quad \sin \theta<0 $$

Step-by-Step Solution

Answer

\( \theta \) lies in Quadrant IV.

Step 1: Understanding Trigonometric Signs by Quadrant

Recall that each quadrant in the Cartesian plane affects the sign of trigonometric functions: \( \tan \theta \) is positive in Quadrants I and III, while \( \sin \theta \) is positive in Quadrants I and II. Therefore, \( \tan \theta < 0 \) occurs in Quadrants II and IV, and \( \sin \theta < 0 \) occurs in Quadrants III and IV.

Step 2: Identifying the Common Quadrant

Since both conditions must be satisfied simultaneously, identify the quadrant where both \( \tan \theta < 0 \) and \( \sin \theta < 0 \). Quadrant IV is the only quadrant where both \( \tan \theta \) and \( \sin \theta \) are negative.

Key Concepts

Trigonometric Functions Signs

Trigonometric functions like sine, cosine, and tangent each have positive and negative signs in different quadrants of the Cartesian plane. Knowing these signs helps us understand the behavior of these functions at specific angles. For sine (\( \sin \theta \)), it is positive when angles are in Quadrants I and II, but negative in Quadrants III and IV. On the other hand, the tangent (\( \tan \theta \)) function is positive when angles are in Quadrants I and III, while it is negative in Quadrants II and IV. This labeling is crucial because it provides clues about the positioning of angles in trigonometric problems, making it easier to solve them. Remembering these signs can help decipher which quadrant an angle or measurement belongs to based on the given trigonometric function values.

Cartesian Plane Quadrants

The Cartesian plane is divided into four quadrants, which are numbered counterclockwise starting from the upper right. Here's a brief overview of each:

• **Quadrant I**: Both the x and y coordinates are positive. This means both sine and cosine are positive, resulting in a positive tangent as well.


• **Quadrant II**: The x coordinate is negative, and y is positive. Sine remains positive, while cosine and tangent are negative.


• **Quadrant III**: Both x and y coordinates are negative. In this quadrant, sine and cosine are negative, but tangent is positive because dividing two negative numbers yields a positive result.


• **Quadrant IV**: The x coordinate is positive, and y is negative. Here, sine is negative, cosine is positive, and tangent is negative.

Understanding these quadrants not only helps in finding trigonometric function signs but also aids in visualizing graphically where specific angle measurements may lie.

Trigonometric Identities Analysis

Trigonometric identities play a pivotal role in simplifying and solving problems involving trigonometric functions. Identities are equations that hold true for all values of the involved variables, provided the variables are within their domains. Some key trigonometric identities include:

• **Pythagorean Identities**: Such as \( \sin^2\theta + \cos^2\theta = 1 \)
• **Reciprocal Identities**: Such as \( \csc\theta = \frac{1}{\sin\theta} \)
• **Quotient Identities**: Such as \( \tan\theta = \frac{\sin\theta}{\cos\theta} \)

These identities are algebraic rules that can be applied to manipulate and understand trigonometric equations. By analyzing these identities, one can determine the relationships between different trigonometric functions, determine restrictions for solutions, and identify clues about which angle or quadrant is involved. Thus, understanding and analyzing trigonometric identities are fundamental in solving trigonometric equations and establishing relationships between the functions involved.