Question

How do you find cos if sin= 5/13?

Answer 1

`cos=+-12/13`

Explanation:

If `sin=5/13`
then
`color(white)("XXX")` the ratio of `("opposite side")/("hypotenuse") = 5/13`

By Pythagorean Theorem
If `"opposite side" = 5 " units"` and `"hypotenuse" = 13 " units"`
`color(white)("XXX")`(for any `"units"`)
then `"adjacent side" = 12 " units"`
and
`cos = ("adjacent side")/("hypotenuse") = 12/13`

However, we need to note that if the angle is in Quadrant II then the `"adjacent side"` will actually be a negative value,
so
`color(white)("XXX")cos=12/13` for an angle in Q I
or
`color(white)("XXX")cos=-12/13` for an angle in Q II
(the angle can't be in Q III or Q IV, since `sin > 0`)

Answer 2

`cos x = +- 12/13`

Explanation:

Another way.
`cos^2 x = 1 - sin^2 x = 1 - 25/169 = 144/169`
`cos x = +- 12/13`
sin x > 0 --> x is either in Quadrant 1 or Quadrant 2.

If x is in Quadrant 1, then, cos x > 0 --> `x = 12/13`
If x is in Quadrant 2, then, cos < 0 --> `cos x = - 12/13`