1) $$ 4 an \left( heta-30^{\circ} ight)=1 $$

Question

1) $$ 4 	an \left(	heta-30^{\circ}ight)=1 $$

UpStudy AI · Accepted Answer

We start with the equation:

$$
4 \tan\left(\theta - 30^\circ\right) = 1.
$$

**Step 1. Isolate the tangent function**

Divide both sides of the equation by 4:

$$
\tan\left(\theta - 30^\circ\right) = \frac{1}{4}.
$$

**Step 2. Solve for $$\theta - 30^\circ$$**

Recall that if $$\tan x = \frac{1}{4}$$, then the general solution for $$x$$ is given by:

$$
x = \arctan\left(\frac{1}{4}\right) + 180^\circ k, \quad k \in \mathbb{Z}.
$$

Here, $$x$$ is $$\theta - 30^\circ$$. So we write:

$$
\theta - 30^\circ = \arctan\left(\frac{1}{4}\right) + 180^\circ k, \quad k \in \mathbb{Z}.
$$

**Step 3. Solve for $$\theta$$**

Add $$30^\circ$$ to both sides:

$$
\theta = 30^\circ + \arctan\left(\frac{1}{4}\right) + 180^\circ k, \quad k \in \mathbb{Z}.
$$

This is the general solution for $$\theta$$.

**Optional: Approximate Numeric Value**

If you need a numerical approximation, note that:

$$
\arctan\left(\frac{1}{4}\right) \approx 14.036^\circ.
$$

Thus, one set of solutions is approximately:

$$
\theta \approx 30^\circ + 14.036^\circ + 180^\circ k = 44.036^\circ + 180^\circ k, \quad k \in \mathbb{Z}.
$$
$$ \theta = 30^\circ + \arctan\left(\frac{1}{4}\right) + 180^\circ k, \quad k \in \mathbb{Z} $$

1) \( 4 \tan \left(\theta-30^{\circ}\right)=1 \)

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