Simplify the following expressions without using a calcutator. 6.1 $$ \frac{\sin 210^{\circ} \cos 300^{\circ} an 240^{\circ}}{\cos 120^{\circ} an 150^{\circ} \sin 330^{\circ}} $$

Question

Simplify the following expressions without using a calcutator. 6.1 $$ \frac{\sin 210^{\circ} \cos 300^{\circ} 	an 240^{\circ}}{\cos 120^{\circ} 	an 150^{\circ} \sin 330^{\circ}} $$

UpStudy AI · Accepted Answer

To simplify the expression:

$$
\frac{\sin 210^{\circ} \cos 300^{\circ} \tan 240^{\circ}}{\cos 120^{\circ} \tan 150^{\circ} \sin 330^{\circ}}
$$

Let's evaluate each trigonometric function step by step:

1. **$$\sin 210^{\circ}$$:**
   - $$210^{\circ} = 180^{\circ} + 30^{\circ}$$ (Third Quadrant)
   - $$\sin 210^{\circ} = -\sin 30^{\circ} = -\frac{1}{2}$$

2. **$$\cos 300^{\circ}$$:**
   - $$300^{\circ} = 360^{\circ} - 60^{\circ}$$ (Fourth Quadrant)
   - $$\cos 300^{\circ} = \cos 60^{\circ} = \frac{1}{2}$$

3. **$$\tan 240^{\circ}$$:**
   - $$240^{\circ} = 180^{\circ} + 60^{\circ}$$ (Third Quadrant)
   - $$\tan 240^{\circ} = \tan 60^{\circ} = \sqrt{3}$$

4. **$$\cos 120^{\circ}$$:**
   - $$120^{\circ} = 180^{\circ} - 60^{\circ}$$ (Second Quadrant)
   - $$\cos 120^{\circ} = -\cos 60^{\circ} = -\frac{1}{2}$$

5. **$$\tan 150^{\circ}$$:**
   - $$150^{\circ} = 180^{\circ} - 30^{\circ}$$ (Second Quadrant)
   - $$\tan 150^{\circ} = -\tan 30^{\circ} = -\frac{\sqrt{3}}{3}$$

6. **$$\sin 330^{\circ}$$:**
   - $$330^{\circ} = 360^{\circ} - 30^{\circ}$$ (Fourth Quadrant)
   - $$\sin 330^{\circ} = -\sin 30^{\circ} = -\frac{1}{2}$$

Now, substitute these values into the original expression:

$$
\frac{\left(-\frac{1}{2}\right) \left(\frac{1}{2}\right) \left(\sqrt{3}\right)}{\left(-\frac{1}{2}\right) \left(-\frac{\sqrt{3}}{3}\right) \left(-\frac{1}{2}\right)} = \frac{-\frac{\sqrt{3}}{4}}{-\frac{\sqrt{3}}{12}} 
$$

Simplify the fraction:

$$
\frac{-\frac{\sqrt{3}}{4}}{-\frac{\sqrt{3}}{12}} = \frac{\sqrt{3}}{4} \times \frac{12}{\sqrt{3}} = \frac{12}{4} = 3
$$

**Final Answer:**  
The expression simplifies to 3.
The expression simplifies to 3.

Simplify the following expressions without using a calcutator.
6.1

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