1. Solve $$ 7 an 2 heta=6 $$ for $$ 0^{\circ} \leq heta

Question

1. Solve $$ 7 	an 2 	heta=6 $$ for $$ 0^{\circ} \leq 	heta<360^{\circ} $$

UpStudy AI · Accepted Answer

To solve the equation $$ 7 \tan 2\theta = 6 $$ for $$ 0^{\circ} \leq \theta < 360^{\circ} $$, follow these steps:

1. **Isolate the Tangent Function:**
   $$
   7 \tan 2\theta = 6 \implies \tan 2\theta = \frac{6}{7}
   $$

2. **Find the General Solution for $$ 2\theta $$:**
   $$
   2\theta = \tan^{-1}\left(\frac{6}{7}\right) + k \cdot 180^{\circ} \quad \text{for integer } k
   $$
   Here, $$ \tan^{-1}\left(\frac{6}{7}\right) $$ is the principal angle whose tangent is $$ \frac{6}{7} $$.

3. **Calculate $$ \tan^{-1}\left(\frac{6}{7}\right) $$ Numerically:**
   $$
   \tan^{-1}\left(\frac{6}{7}\right) \approx 40.62^{\circ}
   $$

4. **Solve for $$ \theta $$:**
   $$
   \theta = \frac{40.62^{\circ} + k \cdot 180^{\circ}}{2} = 20.31^{\circ} + k \cdot 90^{\circ}
   $$
   
5. **Determine All Solutions within the Given Range:**
   We need $$ 0^{\circ} \leq \theta < 360^{\circ} $$. Plugging in integer values for $$ k $$:

- For $$ k = 0 $$: $$ \theta = 20.31^{\circ} $$
   - For $$ k = 1 $$: $$ \theta = 110.31^{\circ} $$
   - For $$ k = 2 $$: $$ \theta = 200.31^{\circ} $$
   - For $$ k = 3 $$: $$ \theta = 290.31^{\circ} $$
   
   $$ k = 4 $$ would yield $$ \theta = 380.31^{\circ} $$, which is outside the desired range.

**Final Solutions:**
$$
\theta \approx 20.31^{\circ}, \quad 110.31^{\circ}, \quad 200.31^{\circ}, \quad \text{and} \quad 290.31^{\circ}
$$

**Rounded to Two Decimal Places:**
$$
\theta \approx 20.31^{\circ},\ 110.31^{\circ},\ 200.31^{\circ},\ \text{and}\ 290.31^{\circ}
$$
The solutions for $$ \theta $$ are approximately $$ 20.31^{\circ} $$, $$ 110.31^{\circ} $$, $$ 200.31^{\circ} $$, and $$ 290.31^{\circ} $$.

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