Question

If sin is of n=3/5, what is the tan of n?

One acute angle of a certain right triangle has measure n. If sin n=3/5, what is the tan of n?

Answer 1

`3/4`

Explanation:

We know from our SOH-CAH-TOA definitions of Trig functions that `sin=(opp)/(hyp)`.

This means that, relative to `n`, `3` is the opposite side, and `5` is the hypotenuse. We know two sides of the triangle, so we can find the third, the adjacent side, with the Pythagorean Theorem

`a^2+b^2=c^2`

We essentially know `a` and `c`, so we can plug them in to get

`3^2+b^2=5^2`

`=>9+b^2=25`

`=>b^2=16`

`=>color(blue)(b=4)`

Relative to `b`, `4` is the adjacent side. Again, leveraging our SOH-CAH-TOA trig definitions, we know

`tan=(opp)/(adj)`

We know `3` is the opposite side, and `4` is the adjacent side, so `tan (n)=3/4`.

Aside:

We know we're dealing with a triangle here with sides of `3` and `5`. You may remember hearing about a

`3-4-5` triangle (where the numbers refer to the side lengths)

We could have leveraged our knowledge of this, deduced that the adjacent side was `4`, plug into our formula for tangent, and find our answer of

`3/4`

Even though the latter was a much more simpler process, I hope you have an appreciation for both methods.

Hope this helps!

Answer 2

`tan n = +- 3/4`

Explanation:

`sin n = 3/5`
n could lie either in Quadrant 1 or Quadrant 2.
`cos^2 n = 1 - sin^2 n = 1 - 9/25 = 16/25`
`cos n = +- 4/5`
Cos n could be either positive or negative because t could be in Q. 1, or Q. 2.
`tan n = sin x/(cos x) = (3/5)(+- 5/4) = +- 3/4`
If n lies in Q. 1, tan n is positive
If n lies in Q. 2, tan n is negative
If n is an acute angle (or arc), n lies in Q. 1, and tan n is positive