1. The problem asks to find the quadrant of the angle \(\theta\) based on the signs of sine, cosine, tangent, secant, and cosecant functions. 2. Recall the quadrant rules for trigonometric functions: - Quadrant I: \(\sin \theta > 0, \cos \theta > 0, \tan \theta > 0\) - Quadrant II: \(\sin \theta > 0, \cos \theta < 0, \tan \theta < 0\) - Quadrant III: \(\sin \theta < 0, \cos \theta < 0, \tan \theta > 0\) - Quadrant IV: \(\sin \theta < 0, \cos \theta > 0, \tan \theta < 0\) 3. Use these rules to determine the quadrant for each problem: - 27. \(\sin \theta > 0, \cos \theta < 0\) means Quadrant II. - 28. \(\sin \theta < 0, \cos \theta > 0\) means Quadrant IV. - 29. \(\sin \theta < 0, \tan \theta < 0\) means Quadrant IV (since \(\tan = \frac{\sin}{\cos}\), if \(\tan < 0\) and \(\sin < 0\), then \(\cos > 0\)). - 30. \(\cos \theta > 0, \tan \theta > 0\) means Quadrant I. - 31. \(\cos \theta > 0, \tan \theta < 0\) means Quadrant IV. - 32. \(\cos \theta < 0, \tan \theta > 0\) means Quadrant III. - 33. \(\sec \theta < 0, \sin \theta > 0\) means Quadrant II (since \(\sec = \frac{1}{\cos}\), \(\sec < 0\) means \(\cos < 0\)). - 34. \(\csc \theta > 0, \cos \theta < 0\) means Quadrant II (since \(\csc = \frac{1}{\sin}\), \(\csc > 0\) means \(\sin > 0\)). Final answers: - 27: Quadrant II - 28: Quadrant IV - 29: Quadrant IV - 30: Quadrant I - 31: Quadrant IV - 32: Quadrant III - 33: Quadrant II - 34: Quadrant II