Problem 35 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

Quadrant IV.

Step 1: Understanding Secant and Tangent

The secant function, which is the reciprocal of cosine, is positive in Quadrants I and IV. The tangent function is negative in Quadrants II and IV.

Step 2: Identify Common Quadrant

Since both \( an \theta < 0\) and \( ext{sec} \theta > 0\) can only be true in Quadrant IV, \theta\ must lie in this quadrant.

Key Concepts

Quadrant Identification

In the unit circle, each angle is divided into four quadrants. Each quadrant has unique properties that help dictate the sign of trigonometric functions such as sine, cosine, tangent, and their reciprocals.

• Quadrant I: All trigonometric functions are positive.
• Quadrant II: Sine is positive, but cosine and tangent are negative.
• Quadrant III: Tangent is positive, while sine and cosine are negative.
• Quadrant IV: Cosine is positive, whereas sine and tangent are negative.

When tasked with identifying the quadrant in which an angle lies, it's essential to consider the signs of the specific trigonometric functions mentioned. By analyzing the conditions presented in a problem, you can deduce the possible quadrant where the angle exists. This process requires closely examining which quadrants meet the conditions provided for each function.

Tangent Function

The tangent function (\(\tan \theta\)) is the ratio of the sine and cosine of an angle: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).This function is periodic and repeats every \(180^\circ\), or \(\pi\) radians. The tangent is positive in:

• Quadrant I: where both sine and cosine are positive.
• Quadrant III: where both sine and cosine are negative (resulting in a positive ratio).

The tangent is negative in:

• Quadrant II: where sine is positive and cosine is negative.
• Quadrant IV: where cosine is positive and sine is negative.

So, when determining where \(\tan \theta < 0\), we focus on Quadrants II and IV, as these are the locations where the sine and cosine have different signs, resulting in a negative tangent value.

Secant Function

The secant function (\(\sec \theta\)) is the reciprocal of the cosine function: \(\sec \theta = \frac{1}{\cos \theta}\).Secant shares many characteristics with cosine, in terms of positivity and negativity in the quadrants:

• Positive in Quadrants: I and IV.
• Negative in Quadrants: II and III.

When \(\sec \theta > 0\), it means \(\cos \theta > 0\), which occurs precisely in Quadrants I and IV. This indicates that these are the quadrants to consider when the secant is defined as positive, allowing us to narrow down where the angle might lie, especially when paired with other function conditions, like a negative tangent.