Problem 107 - Algebra and Trigonometry

Question

If \(\cos \theta>0\) and \(\tan \theta<0,\) explain how to find the quadrant in which \(\theta\) lies.

Step-by-Step Solution

Answer

The angle \(\ theta \) lies in the fourth quadrant.

Step 1: Recalling Trigonometric Properties

It's important to recall that \(\cos \theta\) refers to the x-coordinate and \(\tan \theta\) refers to the ratio of the y-coordinate to the x-coordinate on the unit circle. The sign of these functions can help identify the quadrant of \(\ theta \).

Step 2: Identifying Quadrants from Trigonometric Signs

In the first quadrant, both \(\cos \theta \) and \(\tan \theta \) are positive as both x and y coordinates are positive. In the second quadrant, \(\cos \theta \) is negative and \(\tan \theta \) is positive because the x-coordinate is negative and y-coordinate is positive. In the third quadrant, both \(\cos \theta \) and \(\tan \theta \) are negative as both x and y coordinates are negative. Finally, in the fourth quadrant, \(\cos \theta \) is positive while \(\tan \theta \) is negative because the x-coordinate is positive and the y-coordinate is negative.

Step 3: Applying the Information to the Given Problem

Since the \(\cos \theta \) is positive and \(\tan \theta \) is negative, the angle \(\ theta \) lies in the fourth quadrant because that is where the cosine is positive and the tangent is negative.