2.4 Solve for $$ \alpha $$ if : $$ 2 an \alpha=4 $$ and $$ \alpha \in\left[90^{\circ} ; 360^{\circ} ight] $$

Question

2.4 Solve for $$ \alpha $$ if : $$ 2 	an \alpha=4 $$ and $$ \alpha \in\left[90^{\circ} ; 360^{\circ}ight] $$

UpStudy AI · Accepted Answer

We start with the equation

$$
2 \tan \alpha = 4.
$$

**Step 1. Isolate $$\tan \alpha$$:**

Divide both sides by 2:

$$
\tan \alpha = \frac{4}{2} = 2.
$$

**Step 2. Write the general solution:**

For the tangent function, the general solution is

$$
\alpha = \arctan(2) + k \cdot 180^{\circ}, \quad k \in \mathbb{Z}.
$$

**Step 3. Determine the value of $$\arctan(2)$$:**

Using the inverse tangent,

$$
\arctan(2) \approx 63.4349^{\circ}.
$$

**Step 4. Find solutions within the interval $$\left[90^{\circ}, 360^{\circ}\right]$$:**

- For $$k = 0$$:

$$
\alpha \approx 63.4349^{\circ},
$$

which is not within the interval $$\left[90^{\circ}, 360^{\circ}\right]$$.

- For $$k = 1$$:

$$
\alpha \approx 63.4349^{\circ} + 180^{\circ} = 243.4349^{\circ},
$$

which lies within the interval.

- For $$k = -1$$ or $$k = 2$$, the values fall outside the given interval.

**Conclusion:**

The only solution within $$\left[90^{\circ}, 360^{\circ}\right]$$ is

$$
\alpha \approx 243.4349^{\circ}.
$$
The solution within the interval $$[90^\circ, 360^\circ]$$ is $$ \alpha \approx 243.4349^\circ $$.

2.4 Solve for \( \alpha \) if : \( 2 \tan \alpha=4 \) and \( \alpha \in\left[90^{\circ} ; 360^{\circ}\right] \)