Find the exact value of the following expression. tan ^-1 Square root of 3
The Correct Answer and Explanation is:
To find the exact value of tan−1(3)\tan^{-1}(\sqrt{3})tan−1(3), we need to recognize that this represents the angle θ\thetaθ whose tangent is equal to 3\sqrt{3}3. In other words, we are solving for θ\thetaθ in the equation:tan(θ)=3\tan(\theta) = \sqrt{3}tan(θ)=3
Now, let’s recall the standard values of the tangent function for common angles. We know that:tan(π3)=3\tan\left(\frac{\pi}{3}\right) = \sqrt{3}tan(3π)=3
This tells us that the angle θ\thetaθ whose tangent is 3\sqrt{3}3 is π3\frac{\pi}{3}3π radians.
Thus, the exact value of tan−1(3)\tan^{-1}(\sqrt{3})tan−1(3) is:tan−1(3)=π3\tan^{-1}(\sqrt{3}) = \frac{\pi}{3}tan−1(3)=3π
Explanation:
The inverse tangent function, tan−1(x)\tan^{-1}(x)tan−1(x), gives us the angle θ\thetaθ such that tan(θ)=x\tan(\theta) = xtan(θ)=x. In this case, we are asked to find tan−1(3)\tan^{-1}(\sqrt{3})tan−1(3), meaning we are looking for the angle whose tangent is 3\sqrt{3}3. By recognizing the standard angle where the tangent function equals 3\sqrt{3}3, we can quickly identify that π3\frac{\pi}{3}3π is the correct angle.
In radians, π3\frac{\pi}{3}3π is equivalent to 60 degrees, so we could also express the solution as:tan−1(3)=60∘\tan^{-1}(\sqrt{3}) = 60^\circtan−1(3)=60∘
However, in most mathematical contexts, the answer is typically left in radians, so the exact value is π3\frac{\pi}{3}3π radians.
