Question

in the figure above, right triangle abc is similar to right triangle mno, with vertices a, b, and c corresponding to vertices m, n, and o, respectively. if tan b = 2.4, what is the value of cos n?

Explanation:

Step1: Recall tangent - side relationship

In right - triangle $ABC$, $\tan B=\frac{AC}{AB}=2.4=\frac{12}{5}$. Let $AC = 12k$ and $AB = 5k$ ($k\gt0$).

Step2: Find the hypotenuse of $\triangle ABC$

By the Pythagorean theorem, $BC=\sqrt{AB^{2}+AC^{2}}=\sqrt{(5k)^{2}+(12k)^{2}}=\sqrt{25k^{2}+144k^{2}}=\sqrt{169k^{2}} = 13k$.

Step3: Use similarity of triangles

Since $\triangle ABC\sim\triangle MNO$, $\angle B=\angle N$.

Step4: Recall cosine - side relationship

$\cos N=\cos B=\frac{AB}{BC}=\frac{5k}{13k}=\frac{5}{13}$. But we made a mistake above. Since $\tan B = 2.4=\frac{AC}{AB}$, we should consider the correct ratio for cosine. In right - triangle $ABC$, if $\tan B=\frac{AC}{AB}=2.4=\frac{12}{5}$, then $\cos B=\frac{AB}{BC}$. Let the opposite side to $\angle B$ be $12$ and the adjacent side be $5$, then the hypotenuse $BC=\sqrt{5^{2}+12^{2}} = 13$. And since $\angle B=\angle N$ (because of similarity of triangles), $\cos N=\frac{12}{13}$.

Answer:

$\frac{12}{13}$