find the value of arctan sqrt of 3 /3
The Correct Answer and Explanation is:
The value of arctan(33)\arctan\left(\frac{\sqrt{3}}{3}\right)arctan(33) can be found by recognizing that it corresponds to an angle whose tangent is 33\frac{\sqrt{3}}{3}33.
We know that the tangent of standard angles is frequently used in trigonometric calculations. The angle θ\thetaθ in radians or degrees where tan(θ)=33\tan(\theta) = \frac{\sqrt{3}}{3}tan(θ)=33 is a well-known value in trigonometry. Specifically, this is the angle 30∘30^\circ30∘ or π6\frac{\pi}{6}6π radians.
Step-by-step explanation:
- The tangent function is defined as: tan(θ)=oppositeadjacent\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}tan(θ)=adjacentopposite In this case, we are looking for an angle where the ratio of the opposite side to the adjacent side equals 33\frac{\sqrt{3}}{3}33.
- For angles in a 30-60-90 triangle, we know the following:
- The tangent of 30∘30^\circ30∘ (or π6\frac{\pi}{6}6π radians) is: tan(30∘)=13=33\tan(30^\circ) = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}tan(30∘)=31=33
- Therefore, when tan(θ)=33\tan(\theta) = \frac{\sqrt{3}}{3}tan(θ)=33, the angle θ\thetaθ is exactly 30∘30^\circ30∘ or π6\frac{\pi}{6}6π radians.
Conclusion:
arctan(33)=30∘=π6 radians\arctan\left(\frac{\sqrt{3}}{3}\right) = 30^\circ = \frac{\pi}{6} \text{ radians}arctan(33)=30∘=6π radians
This is the correct angle whose tangent equals 33\frac{\sqrt{3}}{3}33.
