Question

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

Answer 1

Given:

`\cos x=(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=(opposite)/(adjacent) \\ \\ \tan x=(24)/(7) \end{gathered}`

Another solution:

We know that:

`\begin{gathered} \cos x=(1)/(\sec x)=(7)/(25) \\ \\ \sec x=(25)/(7) \end{gathered}`

using the Pythagorean trig identity

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

So, the answer will be:

`\tan x=(24)/(7)`