find the value of Tan-1 square root 3/3
The Correct Answer and Explanation is:
The given expression is tan⁻¹(√3/3), which represents the inverse tangent or arctangent of √3/3.
Step 1: Recognizing the Ratio
We are looking for an angle whose tangent equals √3/3. We can rewrite this in terms of a right triangle to understand it better. Recall that the tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side.
If tan(θ) = √3/3, then the opposite side of the triangle is √3, and the adjacent side is 3.
Step 2: Identify the Angle
The tangent of an angle is equal to √3/3 at a specific standard angle. We can compare the given ratio with the known values for the tangent of standard angles in trigonometry. In fact, the tangent of 30° or π/6 radians is 1/√3, which is equivalent to √3/3 when simplified.
Thus, the angle θ whose tangent is √3/3 is 30° or π/6 radians.
Step 3: Conclusion
Therefore, the value of tan⁻¹(√3/3) is 30° or π/6 radians.
Explanation in Context
Inverse trigonometric functions are used to find angles from known ratios. Here, we used the fact that the inverse tangent function, tan⁻¹, returns an angle whose tangent equals the given ratio. By recognizing √3/3 as the ratio of tangent at 30° or π/6 radians, we were able to conclude that the inverse tangent of √3/3 is 30° or π/6.
In trigonometric terms:
- tan(30°) = √3/3
- tan⁻¹(√3/3) = 30°
The inverse tangent function returns the principal value (usually between -90° and 90°, or -π/2 and π/2 radians) corresponding to the given ratio, making 30° the correct solution.
