Identify the quadrant (or possible quadrants) of an angle θ that satisfies the given conditions. See Example 3. csc θ > 0 , cot θ > 0

Verified step by step guidance

1. Recall that \( \csc \theta = \frac{1}{\sin \theta} \), so \( \csc \theta > 0 \) implies \( \sin \theta > 0 \).
2. Recall that \( \cot \theta = \frac{\cos \theta}{\sin \theta} \), so \( \cot \theta > 0 \) implies \( \cos \theta > 0 \) when \( \sin \theta > 0 \).
3. The sine function is positive in the first and second quadrants.
4. The cosine function is positive in the first and fourth quadrants.
5. Combine the conditions: \( \sin \theta > 0 \) and \( \cos \theta > 0 \) are both true in the first quadrant.

Key Concepts

Cosecant Function

The cosecant function, denoted as csc(θ), is the reciprocal of the sine function. It is defined as csc(θ) = 1/sin(θ). For csc(θ) to be greater than zero, sin(θ) must also be positive, which occurs in the first and second quadrants of the unit circle.

Cotangent Function

The cotangent function, represented as cot(θ), is the reciprocal of the tangent function, defined as cot(θ) = cos(θ)/sin(θ). For cot(θ) to be greater than zero, both sine and cosine must have the same sign, which occurs in the first and third quadrants.

Quadrants of the Unit Circle

The unit circle is divided into four quadrants, each corresponding to specific signs of sine and cosine. In the first quadrant, both sine and cosine are positive; in the second, sine is positive and cosine is negative; in the third, both are negative; and in the fourth, sine is negative and cosine is positive. Understanding these quadrants is essential for determining where the given conditions hold true.