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 the definitions and signs of the trigonometric functions in each quadrant. The cosecant function, \(\csc \theta\), is the reciprocal of sine, so \(\csc \theta = \frac{1}{\sin \theta}\). Therefore, \(\csc \theta > 0\) means \(\sin \theta > 0\).
2. Determine in which quadrants \(\sin \theta\) is positive. Since sine corresponds to the y-coordinate on the unit circle, \(\sin \theta > 0\) in Quadrant I and Quadrant II.
3. Next, analyze the condition \(\cot \theta > 0\). Recall that \(\cot \theta = \frac{\cos \theta}{\sin \theta}\). For \(\cot \theta\) to be positive, the ratio \(\frac{\cos \theta}{\sin \theta}\) must be positive, meaning \(\cos \theta\) and \(\sin \theta\) have the same sign.
4. Since from step 2, \(\sin \theta > 0\), for \(\cot \theta\) to be positive, \(\cos \theta\) must also be positive. Identify the quadrants where both \(\sin \theta\) and \(\cos \theta\) are positive.
5. Conclude that the angle \(\theta\) must lie in Quadrant I, where both sine and cosine are positive, satisfying both \(\csc \theta > 0\) and \(\cot \theta > 0\).

Key Concepts

Signs of Trigonometric Functions in Quadrants

The signs of sine, cosine, tangent, and their reciprocals vary depending on the quadrant of the angle. For example, sine and cosecant are positive in Quadrants I and II, while cotangent is positive in Quadrants I and III. Understanding these sign patterns helps determine the possible quadrants for an angle.

Reciprocal Trigonometric Functions

Cosecant (csc) is the reciprocal of sine (sin), and cotangent (cot) is the reciprocal of tangent (tan). Knowing these relationships allows you to infer the sign of sine and tangent from the signs of cosecant and cotangent, which is essential for identifying the correct quadrant.

Quadrant Determination Using Multiple Conditions

When given multiple inequalities involving trigonometric functions, the solution involves finding the intersection of quadrants where all conditions hold true. This requires combining the sign information of each function to narrow down the possible quadrants for the angle θ.