Find the values of
`(i) " sin "405^(@) " " (ii) " sec " (-1470^(@)) " "(iii) " tan " (-300^(@))`
`(iv) " cot " (585^(@)) " "(v) " cosec " (-750^(@)) " "(vi) " cos " (-2220^(@))`

Answer

To find the values of the given trigonometric functions, we will simplify each angle using periodic properties of trigonometric functions.

### Step-by-Step Solution:

**(i) Find \( \sin 405^\circ \)**

1. **Reduce the angle**:
\[ 405^\circ = 360^\circ + 45^\circ \]
This means \( \sin 405^\circ = \sin(360^\circ + 45^\circ) \).

2. **Use the periodic property**:
\[ \sin(360^\circ + \theta) = \sin \theta \]
Therefore,
\[ \sin 405^\circ = \sin 45^\circ \]

3. **Find the value**:
\[ \sin 45^\circ = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \]

**Final Answer**: \( \sin 405^\circ = \frac{\sqrt{2}}{2} \)

---

**(ii) Find \( \sec(-1470^\circ) \)**

1. **Reduce the angle**:
\[ -1470^\circ = -1440^\circ - 30^\circ \]
Since \( -1440^\circ \) is equivalent to \( 0^\circ \) (as it completes 4 full cycles of 360°), we have:
\[ \sec(-1470^\circ) = \sec(-30^\circ) \]

2. **Use the property of secant**:
\[ \sec(-\theta) = \sec \theta \]
Therefore,
\[ \sec(-30^\circ) = \sec(30^\circ) \]

3. **Find the value**:
\[ \sec(30^\circ) = \frac{1}{\cos(30^\circ)} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3} \]

**Final Answer**: \( \sec(-1470^\circ) = \frac{2\sqrt{3}}{3} \)

---

**(iii) Find \( \tan(-300^\circ) \)**

1. **Use the property of tangent**:
\[ \tan(-\theta) = -\tan(\theta) \]
Therefore,
\[ \tan(-300^\circ) = -\tan(300^\circ) \]

2. **Reduce the angle**:
\[ 300^\circ = 360^\circ - 60^\circ \]
So,
\[ \tan(300^\circ) = \tan(-60^\circ) = -\tan(60^\circ) \]

3. **Find the value**:
\[ \tan(60^\circ) = \sqrt{3} \]
Thus,
\[ \tan(-300^\circ) = -(-\sqrt{3}) = \sqrt{3} \]

**Final Answer**: \( \tan(-300^\circ) = \sqrt{3} \)

---

**(iv) Find \( \cot(585^\circ) \)**

1. **Reduce the angle**:
\[ 585^\circ = 540^\circ + 45^\circ \]
Therefore,
\[ \cot(585^\circ) = \cot(540^\circ + 45^\circ) \]

2. **Use the periodic property**:
\[ \cot(540^\circ + \theta) = \cot \theta \]
Thus,
\[ \cot(585^\circ) = \cot(45^\circ) \]

3. **Find the value**:
\[ \cot(45^\circ) = 1 \]

**Final Answer**: \( \cot(585^\circ) = 1 \)

---

**(v) Find \( \csc(-750^\circ) \)**

1. **Reduce the angle**:
\[ -750^\circ = -720^\circ - 30^\circ \]
Since \( -720^\circ \) is equivalent to \( 0^\circ \), we have:
\[ \csc(-750^\circ) = \csc(-30^\circ) \]

2. **Use the property of cosecant**:
\[ \csc(-\theta) = -\csc(\theta) \]
Therefore,
\[ \csc(-30^\circ) = -\csc(30^\circ) \]

3. **Find the value**:
\[ \csc(30^\circ) = \frac{1}{\sin(30^\circ)} = \frac{1}{\frac{1}{2}} = 2 \]
Thus,
\[ \csc(-750^\circ) = -2 \]

**Final Answer**: \( \csc(-750^\circ) = -2 \)

---

**(vi) Find \( \cos(-2220^\circ) \)**

1. **Reduce the angle**:
\[ -2220^\circ = -2160^\circ - 60^\circ \]
Since \( -2160^\circ \) is equivalent to \( 0^\circ \), we have:
\[ \cos(-2220^\circ) = \cos(-60^\circ) \]

2. **Use the property of cosine**:
\[ \cos(-\theta) = \cos(\theta) \]
Therefore,
\[ \cos(-60^\circ) = \cos(60^\circ) \]

3. **Find the value**:
\[ \cos(60^\circ) = \frac{1}{2} \]

**Final Answer**: \( \cos(-2220^\circ) = \frac{1}{2} \)

---