Evaluate the expressions without using a calculato $$ 1 \quad 4 an 330^{\circ} \cos 135^{\circ} \sin 300^{\circ} $$ $$ 2 \quad \frac{3 an 315^{\circ} \sin 225^{\circ} \sin 120^{\circ}}{\cos 315^{\circ} \cos 210^{\circ}} $$

Question

Evaluate the expressions without using a calculato $$ 1 \quad 4 	an 330^{\circ} \cos 135^{\circ} \sin 300^{\circ} $$ $$ 2 \quad \frac{3 	an 315^{\circ} \sin 225^{\circ} \sin 120^{\circ}}{\cos 315^{\circ} \cos 210^{\circ}} $$

UpStudy AI · Accepted Answer

We start by evaluating each trigonometric function in the expressions.

### Expression 1

Evaluate
$$
4 \tan 330^{\circ} \cos 135^{\circ} \sin 300^{\circ}.
$$

1. **Find $$\tan 330^{\circ}$$:**  
   Since $$330^{\circ} = 360^{\circ} - 30^{\circ}$$, we have
   $$
   \tan 330^{\circ} = -\tan 30^{\circ} = -\frac{1}{\sqrt{3}}.
   $$

2. **Find $$\cos 135^{\circ}$$:**  
   Note that $$135^{\circ} = 180^{\circ} - 45^{\circ}$$, so
   $$
   \cos 135^{\circ} = -\cos 45^{\circ} = -\frac{\sqrt{2}}{2}.
   $$

3. **Find $$\sin 300^{\circ}$$:**  
   Since $$300^{\circ} = 360^{\circ} - 60^{\circ}$$, we get
   $$
   \sin 300^{\circ} = -\sin 60^{\circ} = -\frac{\sqrt{3}}{2}.
   $$

Substitute these into the expression:
$$
4 \cdot \left(-\frac{1}{\sqrt{3}}\right) \cdot \left(-\frac{\sqrt{2}}{2}\right) \cdot \left(-\frac{\sqrt{3}}{2}\right).
$$

Determine the sign:  
$$- \times - \times - = -$$.

Now compute the numerical part:
$$
4 \cdot \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}.
$$
Notice that
$$
\frac{1}{\sqrt{3}} \cdot \sqrt{3} = 1.
$$
Thus, we have:
$$
4 \cdot \frac{\sqrt{2}}{2 \cdot 2} = 4 \cdot \frac{\sqrt{2}}{4} = \sqrt{2}.
$$

Incorporating the negative sign:
$$
-\sqrt{2}.
$$

### Expression 2

Evaluate
$$
\frac{3 \tan 315^{\circ} \sin 225^{\circ} \sin 120^{\circ}}{\cos 315^{\circ} \cos 210^{\circ}}.
$$

1. **Find $$\tan 315^{\circ}$$:**  
   Since $$315^{\circ} = 360^{\circ} - 45^{\circ}$$,
   $$
   \tan 315^{\circ} = -\tan 45^{\circ} = -1.
   $$

2. **Find $$\sin 225^{\circ}$$:**  
   With $$225^{\circ} = 180^{\circ} + 45^{\circ}$$,
   $$
   \sin 225^{\circ} = -\sin 45^{\circ} = -\frac{\sqrt{2}}{2}.
   $$

3. **Find $$\sin 120^{\circ}$$:**  
   Since $$120^{\circ} = 180^{\circ} - 60^{\circ}$$,
   $$
   \sin 120^{\circ} = \sin 60^{\circ} = \frac{\sqrt{3}}{2}.
   $$

4. **Find $$\cos 315^{\circ}$$:**  
   For $$315^{\circ} = 360^{\circ} - 45^{\circ}$$,
   $$
   \cos 315^{\circ} = \cos 45^{\circ} = \frac{\sqrt{2}}{2}.
   $$

5. **Find $$\cos 210^{\circ}$$:**  
   Since $$210^{\circ} = 180^{\circ} + 30^{\circ}$$,
   $$
   \cos 210^{\circ} = -\cos 30^{\circ} = -\frac{\sqrt{3}}{2}.
   $$

Substitute these into the expression:
$$
\frac{3 \cdot (-1) \cdot \left(-\frac{\sqrt{2}}{2}\right) \cdot \left(\frac{\sqrt{3}}{2}\right)}{\left(\frac{\sqrt{2}}{2}\right) \cdot \left(-\frac{\sqrt{3}}{2}\right)}.
$$

**Simplify the numerator:**
$$
3 \cdot (-1) \cdot \left(-\frac{\sqrt{2}}{2}\right) = 3 \cdot \frac{\sqrt{2}}{2},
$$
then multiplying by $$\frac{\sqrt{3}}{2}$$ gives:
$$
3 \cdot \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} = \frac{3 \sqrt{6}}{4}.
$$

**Simplify the denominator:**
$$
\frac{\sqrt{2}}{2} \cdot \left(-\frac{\sqrt{3}}{2}\right) = -\frac{\sqrt{6}}{4}.
$$

Thus, the expression becomes:
$$
\frac{\frac{3 \sqrt{6}}{4}}{-\frac{\sqrt{6}}{4}} = -3.
$$

### Final Answers

$$
\boxed{-\sqrt{2}} \quad \text{and} \quad \boxed{-3}.
$$
$$ \boxed{-\sqrt{2} \text{ and } -3} $$

Evaluate the expressions without using a calculato \( 1 \quad 4 \tan 330^{\circ} \cos 135^{\circ} \sin 300^{\circ} \) \( 2 \quad \frac{3 \tan 315^{\circ} \sin 225^{\circ} \sin 120^{\circ}}{\cos 315^{\circ} \cos 210^{\circ}} \)

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