$$ 4 \sin 2 heta-3 \cos 2 heta=0 $$ Show that $$ an 2 heta=\frac{3}{4} $$ Hence, solve for $$ heta $$ where $$ heta \in\left(-360^{\circ} ;-90^{\circ} ight) $$

Question

$$ 4 \sin 2 	heta-3 \cos 2 	heta=0 $$ Show that $$ 	an 2 	heta=\frac{3}{4} $$ Hence, solve for $$ 	heta $$ where $$ 	heta \in\left(-360^{\circ} ;-90^{\circ}ight) $$

UpStudy AI · Accepted Answer

To solve the equation $$ 4 \sin 2\theta - 3 \cos 2\theta = 0 $$, we can rearrange it to express $$ \tan 2\theta $$.

1. **Rearranging the equation:**
   $$
   4 \sin 2\theta = 3 \cos 2\theta
   $$
   Dividing both sides by $$ \cos 2\theta $$ (assuming $$ \cos 2\theta \neq 0 $$):
   $$
   \frac{4 \sin 2\theta}{\cos 2\theta} = 3
   $$
   This simplifies to:
   $$
   \tan 2\theta = \frac{3}{4}
   $$

Now we have shown that $$ \tan 2\theta = \frac{3}{4} $$.

2. **Solving for $$ \theta $$:**
   We need to find $$ 2\theta $$ such that:
   $$
   2\theta = \tan^{-1}\left(\frac{3}{4}\right) + n \cdot 180^\circ
   $$
   where $$ n $$ is any integer.

Let's calculate $$ \tan^{-1}\left(\frac{3}{4}\right) $$.

Using a calculator or trigonometric tables, we find:
   $$
   2\theta \approx 36.87^\circ
   $$

Therefore, we can express $$ 2\theta $$ as:
   $$
   2\theta \approx 36.87^\circ + n \cdot 180^\circ
   $$

Now, we will consider the values of $$ n $$ that keep $$ \theta $$ within the range $$ (-360^\circ, -90^\circ) $$.

- For $$ n = 0 $$:
     $$
     2\theta \approx 36.87^\circ \quad \Rightarrow \quad \theta \approx 18.435^\circ \quad \text{(not in range)}
     $$

- For $$ n = -1 $$:
     $$
     2\theta \approx 36.87^\circ - 180^\circ \approx -143.13^\circ \quad \Rightarrow \quad \theta \approx -71.565^\circ \quad \text{(in range)}
     $$

- For $$ n = -2 $$:
     $$
     2\theta \approx 36.87^\circ - 360^\circ \approx -323.13^\circ \quad \Rightarrow \quad \theta \approx -161.565^\circ \quad \text{(not in range)}
     $$

Thus, the only solution for $$ \theta $$ in the range $$ (-360^\circ, -90^\circ) $$ is:
$$
\theta \approx -71.565^\circ
$$

In conclusion, the solution is:
$$
\theta \approx -71.57^\circ
$$ (rounded to two decimal places).
$$ \theta \approx -71.57^\circ $$

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