Problem 36 - Precalculus Mathematics for Calculus

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 the fourth quadrant.

Step 1: Understanding Quadrants

The coordinate plane is divided into four quadrants. In the first quadrant, both sine and tangent are positive. In the second quadrant, sine is positive while tangent is negative. In the third quadrant, both sine and tangent are negative. In the fourth quadrant, sine is negative while tangent is positive.

Step 2: Analyzing Given Conditions

We know that \( an \theta < 0\) and \( an \theta\) is negative in the second and fourth quadrants. We also know \( an heta\) measures the ratio of sine over cosine. Therefore, \( an heta < 0\) if either sine or cosine is negative but not both.

Step 3: Checking Sine Condition

Since \( an \theta<0\) is satisfied in both the second and fourth quadrants, let's consider \( ext{sin } \theta <0\). The sine function is negative in the third and fourth quadrants.

Step 4: Finding Common Conditions

Consider where \( an \theta < 0\) and \( ext{sin } \theta < 0\) overlap. Since \( an \theta<0\) in the second and fourth quadrants, and \( ext{sin } \theta < 0\) in the third and fourth quadrants, \( \theta \) must lie in the fourth quadrant where both conditions are satisfied.

Key Concepts

Trigonometric Functions

Trigonometric functions are the building blocks for understanding angles and their relationships in a coordinate plane. There are six main trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent. Each function relates to an angle in a right triangle or the unit circle. - **Sine (sin)**: Measures the ratio of the opposite side to the hypotenuse in a right triangle. - **Cosine (cos)**: Measures the ratio of the adjacent side to the hypotenuse. - **Tangent (tan)**: Measures the ratio of the opposite side to the adjacent side. In the context of trigonometric quadrants, it's important to remember: - The **first quadrant**: All trigonometric functions are positive. - The **second quadrant**: Sine is positive, but tangent and cosine are not. - The **third quadrant**: Tangent is positive, but sine and cosine are negative. - The **fourth quadrant**: Cosine is positive, and sine is negative, but tangent is negative. Understanding where each function is positive or negative helps in solving problems regarding the position of angles in specific quadrants.

Sine Function

The sine function, denoted as \( \sin \theta \), is a fundamental trigonometric function that represents the y-coordinate of a point on the unit circle. It tells us how far up or down a point is from the horizontal axis, which is crucial when determining the sign of \( \sin \theta \) in different quadrants. - In the **first quadrant**, sine values are positive.- In the **second quadrant**, sine remains positive.- It becomes negative in the **third quadrant** because it moves below the x-axis.- It is also negative in the **fourth quadrant**.When solving problems like the one in the exercise, where \( \sin \theta < 0 \), we're looking at the quadrants where sine is negative - the third and fourth. This insight helps narrow down the possibilities when combined with other conditions, such as those given for the tangent function in the problem.

Tangent Function

The tangent function, represented as \( \tan \theta \), is defined as the ratio of sine to cosine: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). This means its value depends on both sine and cosine. Understanding the tangent function is essential for determining its sign across different quadrants:- It is positive in the **first quadrant**, as both sine and cosine are positive.- It is negative in the **second quadrant**, where sine is positive and cosine is negative.- It becomes positive again in the **third quadrant**, where both sine and cosine are negative (negative divided by negative is positive).- Finally, it is negative in the **fourth quadrant**, where sine is negative and cosine is positive.In the problem, you're given \( \tan \theta < 0 \), indicating the second or fourth quadrant, where tangent is negative. By combining this with \( \sin \theta < 0 \) (which indicates the third or fourth quadrants), we can find that both conditions are satisfied only in the fourth quadrant. Understanding these relationships is crucial for navigating problems involving trigonometric identities and quadrants.