Question

Name the quadrant in which the angle \(\theta\) lies. $$ \csc \theta>0, \quad \cot \theta<0 $$

Short answer

The angle \( \theta \) lies in the second quadrant.

Step-by-step solution

Understanding the Given Information

Given that \( \csc \theta > 0 \) and \( \cot \theta < 0 \). The \( \csc \) function is positive when \( \sin \theta \) is positive. Similarly, the \( \cot \) function is negative when the cosine and sine functions have opposite signs.

Determine where \csc \theta > 0

Recall that \( \csc \) function is the reciprocal of \( \sin \). So, if \( \csc \theta > 0 \), \( \sin \theta > 0 \). ewline - \( \sin \) is positive in the first and second quadrants.

Determine where \( \cot \theta < 0 \)

Recall that \( \cot \) function is the ratio of \( \cos \theta \) to \( \sin \theta \). If \( \cot \theta < 0 \), either \( \cos \theta \) is positive and \( \sin \theta \) is negative (or vice versa). ewline Since \( \sin \) is positive (from Step 2), then \$ therefore, \( \cos \theta \) must be negative. \( \cos \) is negative only in the second quadrant.

Conclude the location

Since \( \sin \theta \) is positive and \( \cos \theta \) is negative, the angle \( \theta \) lies in the second quadrant.

Key concepts

Quadrants in Trigonometry

Trigonometry divides the unit circle into four quadrants, each with specific sign rules for sine \(\sin\), cosine \(\cos\), and tangent \(\tan\). The first quadrant (0° to 90°) has both \(\sin \) and \(\cos \) positive. The second quadrant (90° to 180°) has \(\sin \) positive and \(\cos \) negative. In the third quadrant (180° to 270°), both \(\sin \) and \(\cos \) are negative. The fourth quadrant (270° to 360°) has \(\sin \) negative and \(\cos \) positive. To determine the quadrant for a given angle \(\theta \), we look at the provided trigonometric functions and their signs. This way, we identify in which quadrant the angle lies.

Cosecant Function

The cosecant function, denoted as \(\csc \), is the reciprocal of the sine function \(\sin\). This means \(\csc \theta = \frac{1}{\sin \theta}\). Because it's a reciprocal, whenever \(\csc \theta \) is positive, \(\sin \theta \) must also be positive. Therefore, we know that if \(\csc \theta \) is greater than zero, \(\sin \theta \) is positive, which restricts our angle to either the first or second quadrant, where the sine is positive.

Cotangent Function

The cotangent function, denoted as \(\cot \), is the reciprocal of the tangent function and can also be expressed as the ratio of the cosine to the sine function: \(\cot \theta = \frac{\cos \theta}{\sin \theta}\). When \(\cot \theta < 0\), it tells us that \(\cos \theta \) and \(\sin \theta \) have opposite signs. If \(\sin \theta \) is positive (as we know from the positive cosecant), \(\cos \) must be negative. This situation occurs only in the second quadrant.

Sine and Cosine Signs

In trigonometric functions, the signs of sine and cosine vary by quadrant. In the first quadrant, both \(\sin \) and \(\cos \) are positive. In the second quadrant, \(\sin \) is positive and \(\cos \) is negative. The third quadrant has both \(\sin \) and \(\cos \) as negative, and in the fourth quadrant, \(\cos \) is positive while \(\sin \) is negative. Knowing these sign rules helps in determining the correct quadrant for a given trigonometric condition. For example, in our problem, because \(\csc \theta \) is positive (indicating \(\sin \theta \) is positive) and \(\cot \theta \) is negative (meaning \(\cos \theta \) is negative when \(\sin \theta \) is positive), we conclude that the angle \(\theta \) lies in the second quadrant.