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. $$ \csc 60^{\circ}- an 45^{\circ}+\cos 60^{\circ} $$ 4. $$ 3 an 60^{\circ}+\cot 30^{\circ}-2 \cot 45^{\circ} $$ 5. $$ \csc 60^{\circ}-\sec 30^{\circ}+\csc 45^{\circ} $$ 4.

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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. $$ \csc 60^{\circ}-	an 45^{\circ}+\cos 60^{\circ} $$ 4. $$ 3 	an 60^{\circ}+\cot 30^{\circ}-2 \cot 45^{\circ} $$ 5. $$ \csc 60^{\circ}-\sec 30^{\circ}+\csc 45^{\circ} $$ 4.

UpStudy AI · Accepted Answer

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. $$ \sqrt{2} $$
1. 2 2. $$2 + 2\sqrt{3}$$ 3. $$\frac{4 - \sqrt{3}}{2\sqrt{3}}$$ 4. $$4\sqrt{3} - 2$$ 5. $$\sqrt{2}$$

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