find the exact value of sin(tan^(-1)(sqrt(3)/3)) without a calculator. sin(tan^(-1)(sqrt(3)/3)) = □…

Explanation:

Step1: Let $\theta=\tan^{- 1}\frac{\sqrt{3}}{3}$

This means $\tan\theta=\frac{\sqrt{3}}{3}$, and $\theta\in(-\frac{\pi}{2},\frac{\pi}{2})$.

Step2: Recall the right - triangle relationship

Since $\tan\theta=\frac{y}{x}=\frac{\sqrt{3}}{3}$, we can consider a right - triangle where $y = \sqrt{3}$ and $x = 3$. Then, by the Pythagorean theorem $r=\sqrt{x^{2}+y^{2}}=\sqrt{3^{2}+\left(\sqrt{3}\right)^{2}}=\sqrt{9 + 3}=\sqrt{12}=2\sqrt{3}$.

Step3: Find $\sin\theta$

We know that $\sin\theta=\frac{y}{r}$. Substituting $y = \sqrt{3}$ and $r=2\sqrt{3}$, we get $\sin\theta=\frac{\sqrt{3}}{2\sqrt{3}}=\frac{1}{2}$.

Answer:

$\frac{1}{2}$