106. Explain why \( \tan 90^{\circ} \) is undefined. 106. ficos \( \theta>0 \) and \( \tan \theta<0 \), exflain how to find the quadrant in which \( \theta \) lies.

When we look at the unit circle, the tangent function is defined as the ratio of the sine and cosine: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). At \( 90^{\circ} \) (or \( \frac{\pi}{2} \) radians), the cosine of the angle is 0. Since division by zero is undefined in mathematics, this means that \( \tan 90^{\circ} \) results in an undefined value. To find the quadrant where \( \theta \) lies with \( \cos \theta > 0 \) and \( \tan \theta < 0 \), we can use the signs of sine and cosine in each quadrant. Since \( \cos \theta > 0 \), \( \theta \) must be in the first or fourth quadrant. However, \( \tan \theta < 0 \) implies that sine must be negative, which only occurs in the fourth quadrant. Therefore, \( \theta \) is in the fourth quadrant.