TEST YOURSELF: Using the table of values of the trigonometric ratios for \( 30^{\circ}, 45^{\circ} \), and 6 expressions. 1. \( \sec 60^{\circ} \cot 45^{\circ} \) 2. \( \sec 45^{\circ} \csc 45^{\circ}+\sec 60^{\circ} \cot 30^{\circ} \) 3. \( \csc 60^{\circ}-\tan 45^{\circ}+\cos 60^{\circ} \) 4. \( 3 \tan 60^{\circ}+\cot 30^{\circ}-2 \cot 45^{\circ} \) 5. \( \csc 60^{\circ}-\sec 30^{\circ}+\csc 45^{\circ} \) 4.

To solve the expressions involving trigonometric ratios for \(30^{\circ}\), \(45^{\circ}\), and \(60^{\circ}\), we first need to recall the values of the trigonometric functions for these angles: - \( \sin 30^{\circ} = \frac{1}{2} \) - \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \) - \( \tan 30^{\circ} = \frac{1}{\sqrt{3}} \) - \( \csc 30^{\circ} = 2 \) - \( \sec 30^{\circ} = \frac{2}{\sqrt{3}} \) - \( \cot 30^{\circ} = \sqrt{3} \) - \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \) - \( \cos 45^{\circ} = \frac{\sqrt{2}}{2} \) - \( \tan 45^{\circ} = 1 \) - \( \csc 45^{\circ} = \sqrt{2} \) - \( \sec 45^{\circ} = \sqrt{2} \) - \( \cot 45^{\circ} = 1 \) - \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \) - \( \cos 60^{\circ} = \frac{1}{2} \) - \( \tan 60^{\circ} = \sqrt{3} \) - \( \csc 60^{\circ} = \frac{2}{\sqrt{3}} \) - \( \sec 60^{\circ} = 2 \) - \( \cot 60^{\circ} = \frac{1}{\sqrt{3}} \) Now, let's evaluate each expression step by step. ### 1. \( \sec 60^{\circ} \cot 45^{\circ} \) \[ \sec 60^{\circ} = 2, \quad \cot 45^{\circ} = 1 \] \[ \sec 60^{\circ} \cot 45^{\circ} = 2 \cdot 1 = 2 \] ### 2. \( \sec 45^{\circ} \csc 45^{\circ} + \sec 60^{\circ} \cot 30^{\circ} \) \[ \sec 45^{\circ} = \sqrt{2}, \quad \csc 45^{\circ} = \sqrt{2} \] \[ \sec 60^{\circ} = 2, \quad \cot 30^{\circ} = \sqrt{3} \] \[ \sec 45^{\circ} \csc 45^{\circ} = \sqrt{2} \cdot \sqrt{2} = 2 \] \[ \sec 60^{\circ} \cot 30^{\circ} = 2 \cdot \sqrt{3} = 2\sqrt{3} \] \[ \sec 45^{\circ} \csc 45^{\circ} + \sec 60^{\circ} \cot 30^{\circ} = 2 + 2\sqrt{3} \] ### 3. \( \csc 60^{\circ} - \tan 45^{\circ} + \cos 60^{\circ} \) \[ \csc 60^{\circ} = \frac{2}{\sqrt{3}}, \quad \tan 45^{\circ} = 1, \quad \cos 60^{\circ} = \frac{1}{2} \] \[ \csc 60^{\circ} - \tan 45^{\circ} + \cos 60^{\circ} = \frac{2}{\sqrt{3}} - 1 + \frac{1}{2} \] To combine these, we can find a common denominator, which is \(2\sqrt{3}\): \[ = \frac{4}{2\sqrt{3}} - \frac{2\sqrt{3}}{2\sqrt{3}} + \frac{\sqrt{3}}{2\sqrt{3}} = \frac{4 - 2\sqrt{3} + \sqrt{3}}{2\sqrt{3}} = \frac{4 - \sqrt{3}}{2\sqrt{3}} \] ### 4. \( 3 \tan 60^{\circ} + \cot 30^{\circ} - 2 \cot 45^{\circ} \) \[ \tan 60^{\circ} = \sqrt{3}, \quad \cot 30^{\circ} = \sqrt{3}, \quad \cot 45^{\circ} = 1 \] \[ 3 \tan 60^{\circ} = 3\sqrt{3}, \quad \cot 30^{\circ} = \sqrt{3}, \quad -2 \cot 45^{\circ} = -2 \] \[ 3 \tan 60^{\circ} + \cot 30^{\circ} - 2 \cot 45^{\circ} = 3\sqrt{3} + \sqrt{3} - 2 = 4\sqrt{3} - 2 \] ### 5. \( \csc 60^{\circ} - \sec 30^{\circ} + \csc 45^{\circ} \) \[ \csc 60^{\circ} = \frac{2}{\sqrt{3}}, \quad \sec 30^{\circ} = \frac{2}{\sqrt{3}}, \quad \csc 45^{\circ} = \sqrt{2} \] \[ \csc 60^{\circ} - \sec 30^{\circ} + \csc 45^{\circ} = \frac{2}{\sqrt{3}} - \frac{2}{\sqrt{3}} + \sqrt{2} = \sqrt{2} \] ### Summary of Results: 1. \( 2 \) 2. \( 2 + 2\sqrt{3} \) 3. \( \frac{4 - \sqrt{3}}{2\sqrt{3}} \) 4. \( 4\sqrt{3} - 2 \) 5. \( \sqrt{2} \)