Problem 36 - Algebra and Trigonometry

Question

33–36 Find the quadrant in which \(\theta\) lies from the information given. \(\csc \theta>0 \quad\) and \(\quad \cos \theta<0\)

Step-by-Step Solution

Answer

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

Step 1: Analyze the given conditions for cosecant

Since \( \csc \theta > 0 \), we know that \( \sin \theta > 0 \). This implies that \( \theta \) could lie either in the first quadrant where all trigonometric functions are positive, or in the second quadrant where sine is positive, but cosine is negative.

Step 2: Analyze the given condition for cosine

The condition \( \cos \theta < 0 \) tells us that \( \theta \) cannot be in the first quadrant (where cosine is positive). It must lie where cosine is negative.

Step 3: Determine the quadrant using both conditions

By combining both conditions \( \csc \theta > 0 \) (which implies \( \sin \theta > 0 \)) and \( \cos \theta < 0 \), \( \theta \) is confirmed to be in the second quadrant (where sine is positive and cosine is negative).

Key Concepts

Cosecant Function

The cosecant function, represented as \( \csc \theta \), is the reciprocal of the sine function. This means \( \csc \theta = \frac{1}{\sin \theta} \). Because it is the reciprocal, when \( \csc \theta > 0 \), it implies \( \sin \theta > 0 \) as well. Understanding

• Cosecant is undefined for any angle where \( \sin \theta = 0 \) because division by zero is undefined.
• Usually, cosecant appears when dealing with trigonometric ratios for angles in right-angled triangles.
• The function is useful in various mathematical applications including calculus and differential equations.

Although less commonly used than sine, cosine, and tangent, recognizing that \( \csc \theta > 0 \) directly tells us that \( \sin \theta > 0 \) helps solve problems involving trigonometric inequalities.

Cosine Function

The cosine function, indicated by \( \cos \theta \), is a principal trigonometric function. It is crucial in determining the direction and angle relationships. In any circle,

• Cosine corresponds to the horizontal coordinate of a point moving around the circle.
• A positive cosine value indicates the angle lies in either the first or fourth quadrant of the unit circle.
• A negative cosine value means the angle is in either the second or third quadrant.

Therefore, the condition \( \cos \theta < 0 \) suggests that the angle cannot be in the first quadrant and must lie in a section of the unit circle where cosine is negative. This information is vital when solving trigonometry problems, especially those involving angle identification in specific quadrants.

Sine Function

The sine function, denoted as \( \sin \theta \), is fundamental to trigonometry. This function relates to the vertical coordinate of a point on the unit circle. When analyzing

• Positive sine values indicate the angle is in the first or second quadrant, where it aligns with positive vertical positions.
• Negative sine values occur in the third and fourth quadrants.

In context, knowing \( \sin \theta > 0 \) helps determine that the angle \( \theta \) resides where only first quadrant or second quadrant conditions are valid. Such awareness simplifies the evaluation of trigonometric expressions and inequalities, particularly when combined with other function conditions.

Trigonometric Inequalities

Trigonometric inequalities are expressions involving trigonometric functions that indicate ranges of angles meeting certain conditions. They often resemble inequalities such as \( \csc \theta > 0 \) or \( \cos \theta < 0 \). To effectively solve these,

• Understand the sign of the trigonometric function in each quadrant, crucial for determining where an angle might lie.
• Use the reciprocal identities to translate complementary conditions as shown in the relationship between \( \csc \theta \) and \( \sin \theta \).
• Combine information logically, taking both positive and negative function conditions into account.

In our example, by evaluating \( \csc \theta > 0 \) (sine > 0) and \( \cos \theta < 0 \), the inequalities confirm \( \theta \) must be in the second quadrant, demonstrating how trigonometric inequalities guide us in locating angles precisely.