Arctan square root 3 /3
The Correct Answer and Explanation is:
The expression you provided is the inverse tangent of 33\frac{\sqrt{3}}{3}33, written as:arctan(33)\arctan\left(\frac{\sqrt{3}}{3}\right)arctan(33)
To find the exact value, we need to recall some fundamental trigonometric relationships. Specifically, we are looking for an angle θ\thetaθ whose tangent value equals 33\frac{\sqrt{3}}{3}33. That is:tan(θ)=33\tan(\theta) = \frac{\sqrt{3}}{3}tan(θ)=33
From trigonometry, we know that:tan(π6)=33\tan\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{3}tan(6π)=33
Thus, the angle θ=π6\theta = \frac{\pi}{6}θ=6π is the solution to the equation. Therefore:arctan(33)=π6\arctan\left(\frac{\sqrt{3}}{3}\right) = \frac{\pi}{6}arctan(33)=6π
Explanation:
The inverse tangent function, arctan(x)\arctan(x)arctan(x), gives the angle whose tangent is xxx. The tangent function, tan(θ)\tan(\theta)tan(θ), for a standard 30-60-90 right triangle, has known values for the angles:
- At θ=π6\theta = \frac{\pi}{6}θ=6π or 30°, the ratio of the opposite side to the adjacent side is 13\frac{1}{\sqrt{3}}31, or equivalently, 33\frac{\sqrt{3}}{3}33.
- In this case, we are dealing with the ratio 33\frac{\sqrt{3}}{3}33, which directly corresponds to the angle π6\frac{\pi}{6}6π.
So, the solution to the inverse tangent of 33\frac{\sqrt{3}}{3}33 is:arctan(33)=π6 radians\arctan\left(\frac{\sqrt{3}}{3}\right) = \frac{\pi}{6} \text{ radians}arctan(33)=6π radians
This is the simplest and most exact answer in radians. If you were to convert it to degrees, π6\frac{\pi}{6}6π radians equals 30°.
