Question

Fill in Sin, Cos, and tan ratio for angle x.
Sin X = 4/5 (28/35 simplified)

Answer 1

Given:
`\sin(x) = (4/5)`.

Assuming that
`0 < x < 90^(\circ)`,
`\cos(x) = (3/5)` while
`\tan(x) = (4/3)`.

Explanation:

By the Pythagorean identity
`\sin^(2)(x) + \cos^(2)(x) = 1`.

Assuming that
`0 < x < 90^(\circ)`,
`0 < \cos(x) < 1`.

Rearrange the Pythagorean identity to find an expression for
`\cos(x)`.

`\cos^(2)(x) = 1 - \sin^(2)(x)`.

Given that
`0 < \cos(x) < 1`:

`\begin{aligned} &\cos(x) \\ &= \sqrt{1 - \sin^(2)(x)} \\ &= \sqrt{1 - \left((4)/(5)\right)^(2)} \\ &= \sqrt{1 - (16)/(25)} \\ &= (3)/(5)\end{aligned}`.

Hence,
`\tan(x)` would be:

`\begin{aligned}& \tan(x) \\ &= (\sin(x))/(\cos(x)) \\ &= ((4/5))/((3/5)) \\ &= (4)/(3)\end{aligned}`.