Question

If `sin (x) = 3/5,` what is the value of `tan(x)`?

Answer 1

`tan(x) = +-3/4`

Explanation:

This value of `sin(x)` occurs in a `3,4,5` right angled triangle.

Here's a picture I made earlier...

In this triangle we have:

`sin(A) = "opposite"/"hypotenuse" = 3/5`

`cos(A) = "adjacent"/"hypotenuse" = 4/5`

`tan(A) = "opposite"/"adjacent" = 3/4`

Hence if `x` is in Q1 and `sin(x) = 3/5` then `tan(x) = 3/4`

What about other quadrants?

Regardless of which quadrant `x` is in we have:

`cos(x) = +-sqrt(1-sin^2(x))`

In order that `sin(x) > 0` we find that `x` is in Q1 or Q2.

If `x` is in Q2, then `cos(x)` is negative and `tan(x) = sin(x)/cos(x)` is negative too.

So if `x` is in Q2 and `sin(x) = 3/5` then:

`cos(x) = -sqrt(1-sin^2(x)) = -sqrt(1-3^2/5^2) = -sqrt(4^2/5^2) = -4/5`

and:

`tan(x) = sin(x)/cos(x) = (3/5)/(-4/5) = -3/4`