Questions: Determine in which of the four quadrants angle (boldsymboltheta) is located given [ sec (theta)<0 text and tan (theta)<0 ]

Transcript text: Determine in which of the four quadrants angle $\boldsymbol{\theta}$ is located given \[ \sec (\theta)<0 \text { and } \tan (\theta)<0 \]

Solution

Solution Steps

Step 1: Understand the Trigonometric Function Signs in Each Quadrant

In Quadrant I, all trigonometric functions are positive. In Quadrant II, $\sin$ and $\csc$ are positive; $\cos$, $\sec$, $\tan$, and $\cot$ are negative. In Quadrant III, $\tan$ and $\cot$ are positive; $\sin$, $\csc$, $\cos$, and $\sec$ are negative. In Quadrant IV, $\cos$ and $\sec$ are positive; $\sin$, $\csc$, $\tan$, and $\cot$ are negative.

Step 2: Apply the Signs to Determine the Quadrant

Given that $sec(\theta)$ is negative, it implies that $\theta$ could lie in Quadrant(s): 4.