QUESTION 6 Sinplify the following expressions without using a calculator. $$ 6.1 \quad \frac{\sin 210^{\circ} \cos 300^{\circ} an 240^{\circ}}{\cos 120^{\circ} an 150^{\circ} \sin 330^{\circ}} $$ 6.2 $$ [\sin (- heta)+\cos (360- heta)]\left[\cos (90- heta)+\frac{\sin heta}{ an heta} ight] $$ $$ 6.3 \quad $$ If $$ an x=m+\frac{1}{m}, 90^{\circ} \leq x \leq 270^{\circ} $$ and $$ m^{2}+\frac{1}{m^{2}}=1 $$ Calculate the value of $$ x $$ without the use of a calculator.

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QUESTION 6 Sinplify the following expressions without using a calculator. $$ 6.1 \quad \frac{\sin 210^{\circ} \cos 300^{\circ} 	an 240^{\circ}}{\cos 120^{\circ} 	an 150^{\circ} \sin 330^{\circ}} $$ 6.2 $$ [\sin (-	heta)+\cos (360-	heta)]\left[\cos (90-	heta)+\frac{\sin 	heta}{	an 	heta}ight] $$ $$ 6.3 \quad $$ If $$ 	an x=m+\frac{1}{m}, 90^{\circ} \leq x \leq 270^{\circ} $$ and $$ m^{2}+\frac{1}{m^{2}}=1 $$ Calculate the value of $$ x $$ without the use of a calculator.

UpStudy AI · Accepted Answer

Let's tackle each part of the question step-by-step.

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### **6.1 Simplify the Expression**

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

**Step 1: Evaluate Each Trigonometric Function**

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)
   - $$\tan 150^{\circ} = -\tan 30^{\circ} = -\frac{1}{\sqrt{3}}$$

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

**Step 2: Substitute and Simplify**

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

**Answer:**  
$$\boxed{3}$$

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### **6.2 Simplify the Expression**

$$
\left[\sin (-\theta) + \cos (360^{\circ} - \theta)\right] \left[\cos (90^{\circ} - \theta) + \frac{\sin \theta}{\tan \theta}\right]
$$

**Step 1: Simplify Each Component Using Trigonometric Identities**

1. **$$\sin (-\theta)$$**
   - $$\sin (-\theta) = -\sin \theta$$

2. **$$\cos (360^{\circ} - \theta)$$**
   - $$\cos (360^{\circ} - \theta) = \cos \theta$$

So, the first bracket becomes:
   $$
   -\sin \theta + \cos \theta
   $$

3. **$$\cos (90^{\circ} - \theta)$$**
   - $$\cos (90^{\circ} - \theta) = \sin \theta$$

4. **$$\frac{\sin \theta}{\tan \theta}$$**
   - $$\tan \theta = \frac{\sin \theta}{\cos \theta}$$
   - $$\frac{\sin \theta}{\tan \theta} = \frac{\sin \theta}{\frac{\sin \theta}{\cos \theta}} = \cos \theta$$

So, the second bracket becomes:
   $$
   \sin \theta + \cos \theta
   $$

**Step 2: Multiply the Simplified Brackets**

$$
(-\sin \theta + \cos \theta)(\sin \theta + \cos \theta) = \cos^2 \theta - \sin^2 \theta = \cos (2\theta)
$$

**Answer:**  
$$\boxed{\cos 2\theta}$$

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### **6.3 Find the Value of $$x$$**

Given:
$$
\tan x = m + \frac{1}{m}, \quad 90^{\circ} \leq x \leq 270^{\circ}
$$
$$
m^2 + \frac{1}{m^2} = 1
$$

**Step 1: Relate the Given Equations**

$$
\left(m + \frac{1}{m}\right)^2 = m^2 + 2 \times m \times \frac{1}{m} + \frac{1}{m^2} = m^2 + \frac{1}{m^2} + 2 = 3
$$
$$
m + \frac{1}{m} = \pm \sqrt{3}
$$

Thus:
$$
\tan x = \pm \sqrt{3}
$$

**Step 2: Determine Possible Values of $$x$$ in the Given Interval**

- **$$\tan x = \sqrt{3}$$**
  - General solution: $$x = 60^{\circ} + 180^{\circ}k$$, where $$k$$ is an integer.
  - Within $$90^{\circ} \leq x \leq 270^{\circ}$$: $$x = 240^{\circ}$$

- **$$\tan x = -\sqrt{3}$$**
  - General solution: $$x = 120^{\circ} + 180^{\circ}k$$, where $$k$$ is an integer.
  - Within $$90^{\circ} \leq x \leq 270^{\circ}$$: $$x = 120^{\circ}$$

**Answer:**  
$$\boxed{120^{\circ} \text{ and } 240^{\circ}}$$
$$ x = 120^{\circ} $$ or $$ x = 240^{\circ} $$

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