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If sin(x) = four fifths, then cos(x) = three fifths.
Sin-1(0.8) = 53.13 degrees.
Cos(53.13) = 0.6.
There are many ways you could work this out. One of them is to use the relationship between the squares of the sine and cosine:
sin2(x) = 1 - cos2(x)
(4/5)2 = 1 - cos2(x)
16/25 = 1 - cos2(x)
-cos2(x) = - 1 + 16/25
cos2(x) = 1 - 16/25
cos2(x) = 9/25
cos(x) = ±3/5
You could also work it out using good ol' Pythagorean theorem:
Let:
a ≡ adjacent
o ≡ opposite
h ≡ hypotenuse
Then we know:
sine = o/h
cosine = a/h
h2 = a2 + o2
We already have our opposite and hypotenuse sides, as we're given those with the value of the sine. We need to work out the adjacent side then:
h2 = a2 + o2
∴ a2 = h2 - o2
∴ a = (h2 - o2)1/2
∴ a = (52 - 42)1/2
∴ a = (25 - 16)1/2
∴ a = 91/2
∴ a = ±3
And now that we have the length of the adjacent side, as well as the hypotenuse, we know the cosine of the angle as that is is the ratio between the two:
cosine = a/h
∴cosine = ±3/5
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