Question

find the tangent of ∠i. write your answer in simplified, rationalized form. do not round. tan (i) =

Explanation:

Step1: Recall tangent - ratio definition

In a right - triangle, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$ for an acute angle $\theta$. For $\angle I$, the side opposite $\angle I$ is $KJ$ and the side adjacent to $\angle I$ is $IJ$.

Step2: Identify opposite and adjacent sides

Given $KJ$ (opposite to $\angle I$) and $IJ = 4$ (adjacent to $\angle I$), and assume we need to find the length of $KJ$ using the Pythagorean theorem. Let $KJ = x$. By the Pythagorean theorem $x=\sqrt{10^{2}-4^{2}}=\sqrt{100 - 16}=\sqrt{84}=2\sqrt{21}$. But if we consider the sides in terms of the tangent formula with respect to $\angle I$, the opposite side to $\angle I$ is $KJ$ and adjacent is $IJ$. So $\tan(I)=\frac{KJ}{IJ}$. Since $KJ=\sqrt{10^{2}-4^{2}}=\sqrt{84} = 2\sqrt{21}$ and $IJ = 4$, then $\tan(I)=\frac{\sqrt{10^{2}-4^{2}}}{4}=\frac{\sqrt{84}}{4}=\frac{2\sqrt{21}}{4}=\frac{\sqrt{21}}{2}$.

Answer:

$\frac{\sqrt{21}}{2}$