$$ 6.1 \frac{\sin 210^{\circ} \cos 300^{\circ} an 240^{\circ}}{\cos 120^{\circ} an 150^{\circ} \sin 330^{\circ}} $$ $$ 6.2[\sin (- heta)+\cos (360- heta)]\left[\cos (90- heta)+\frac{\sin heta}{ an heta} ight] $$ 6.3 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. C

Question

$$ 6.1 \frac{\sin 210^{\circ} \cos 300^{\circ} 	an 240^{\circ}}{\cos 120^{\circ} 	an 150^{\circ} \sin 330^{\circ}} $$ $$ 6.2[\sin (-	heta)+\cos (360-	heta)]\left[\cos (90-	heta)+\frac{\sin 	heta}{	an 	heta}ight] $$ 6.3 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. C

UpStudy AI · Accepted Answer

To solve the problem:

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

**Step-by-Step Solution:**

1. **Express $$ \tan x $$ in terms of $$ m $$:**
   $$
   \tan x = m + \frac{1}{m}
   $$

2. **Use the given equation $$ m^{2} + \frac{1}{m^{2}} = 1 $$:**
   $$
   m^{2} + \frac{1}{m^{2}} = 1
   $$
   
   Notice that:
   $$
   \left( m + \frac{1}{m} \right)^2 = m^2 + 2 \cdot m \cdot \frac{1}{m} + \frac{1}{m^2} = m^2 + 2 + \frac{1}{m^2}
   $$
   
   Therefore:
   $$
   \left( m + \frac{1}{m} \right)^2 = 1 + 2 = 3 \implies m + \frac{1}{m} = \pm\sqrt{3}
   $$
   
   So:
   $$
   \tan x = \pm\sqrt{3}
   $$

3. **Determine $$ x $$ within the given interval $$ 90^{\circ} \leq x \leq 270^{\circ} $$:**

- **If $$ \tan x = \sqrt{3} $$:**
     - The general solution for $$ \tan x = \sqrt{3} $$ is $$ x = 60^{\circ} + k \cdot 180^{\circ} $$.
     - Within $$ 90^{\circ} \leq x \leq 270^{\circ} $$:
       $$
       x = 240^{\circ}
       $$
   
   - **If $$ \tan x = -\sqrt{3} $$:**
     - The general solution for $$ \tan x = -\sqrt{3} $$ is $$ x = 120^{\circ} + k \cdot 180^{\circ} $$.
     - Within $$ 90^{\circ} \leq x \leq 270^{\circ} $$:
       $$
       x = 120^{\circ}
       $$
   
4. **Conclusion:**
   - The possible values of $$ x $$ within the specified interval that satisfy the given conditions are:
     $$
     x = 120^{\circ} \quad \text{and} \quad x = 240^{\circ}
     $$

**Final Answer:**
The possible values of $$ x $$ are $$120^\circ$$ and $$240^\circ$$.
The possible values of $$ x $$ are $$120^\circ$$ and $$240^\circ$$.

6.3 If and
Calculate the value of without the use of a calculator.
C

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6.3 If and Calculate the value of without the use of a calculator. C