Question

use the Pythagorean trig identity to find tanx if cosx equals 7/25

Answer 1

Given:

$\cos x=\frac{7}{25}$

we know that:

$\cos x=\frac{adjacent}{\text{hypotenuse}}$

So, the adjacent side to x = 7 and the hypotenuse = 25

We can calculate the opposite side by the Pythagorean theorem

So, the opposite side will be:

$\sqrt[]{25^2-7^2}=\sqrt[]{625-49}=\sqrt[]{576}=24$

So, tan x will be:

$\begin{gathered} \tan x=\frac{opposite}{adjacent} \\ \\ \tan x=\frac{24}{7} \end{gathered}$

Another solution:

We know that:

$\begin{gathered} \cos x=\frac{1}{\sec x}=\frac{7}{25} \\ \\ \sec x=\frac{25}{7} \end{gathered}$

using the Pythagorean trig identity

$\begin{gathered} \tan ^2x+1=\sec ^2x \\ \tan ^2x=\sec ^2x-1=(\frac{25}{7})^2-1=\frac{625}{49}-1 \\ \\ \tan ^2x=\frac{576}{49} \\ \\ \tan x=\sqrt[]{\frac{576}{49}}=\frac{24}{7} \end{gathered}$

So, the answer will be:

$\tan x=\frac{24}{7}$