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Answer 1
Thomas Walker
To solve the problem, we need to find all angles $\theta$ such that $\tan(\theta) = 3$ within the interval $[0, 2\pi]$.
Step 1: Recognize that $\tan(\theta)$ is positive in the first and third quadrants.
Step 2: The reference angle $\alpha$ for $\tan(\alpha) = 3$ is found using $\alpha = \arctan(3)$.
Step 3: Calculate $\alpha$:
$\alpha = \arctan(3) \approx 1.249$ radians.
Step 4: Identify the angles in the first and third quadrants:
$\theta_1 = \alpha = \arctan(3) \approx 1.249$ radians
$\theta_2 = \pi + \alpha = \pi + \arctan(3) \approx 4.391$ radians.
Therefore, the solutions are $\theta \approx 1.249$ radians and $\theta \approx 4.391$ radians.
Answer 2
James Taylor
We start by noting that $ an( heta) = 3$ implies $ heta$ is in the first or third quadrant since $ an( heta)$ is positive in these quadrants.
Step 1: Compute the principal value of $ heta$:
$ heta_{p} = arctan(3) approx 1.249$ radians.
Step 2: Determine the equivalent angle in the third quadrant:
$ heta = pi + heta_{p} = pi + 1.249 approx 4.391$ radians.
Thus, the angles satisfying $ an( heta) = 3$ in the interval $[0, 2pi]$ are $ heta approx 1.249$ radians and $ heta approx 4.391$ radians.
Answer 3
William King
Since $ an( heta) = 3$, $ heta$ must be in the first or third quadrant.
Find the reference angle $alpha$:
$alpha = arctan(3) approx 1.249$ radians.
The solutions are:
$ heta_1 = alpha approx 1.249$ radians
$ heta_2 = pi + alpha approx 4.391$ radians.
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