Question

Evaluate using Pythagorean identities?

Find `sintheta` and `costheta` if `tantheta`=`1/5` and `sintheta`>0

Answer 1

`sintheta=1/sqrt(26)` and `costheta=5/sqrt(26)`

Explanation:

Note: Here, I take `x=theta`.

The Pythagorean identities are:

`sin^2x+cos^2x=1`

`tan^2x+1=sec^2x`

`1+cot^2x=csc^2x`

We'll use equation 2 first:

`tan^2x+1=sec^2x`

Here, `tanx=1/5` and so `tan^2x=1/25`. Inputting that into the earlier formula:

`sec^2x=1/25+1=26/25`, so:

`secx=sqrt(26)/5`, and as `secx=1/cosx`, we know that:

`color(red)(cosx=5/sqrt(26))`

Now we use equation 1:

`sin^2x+cos^2x=1`

So, we can say that:

`sin^2x=1-cos^2x`. Here, `cosx=5/sqrt(26)` and so `cos^2x=25/26`. Inputting:

`sin^2x=1-25/26`

`sin^2x=1/26`

`color(red)(sinx=1/sqrt(26))`

Answer 2

`cos t = 5/sqrt26`
`sin t = 1/sqrt26`

Explanation:

`tan t = 1/5`
`cos^2 t = 1/(1 + tan^2 t) = 1/(1 + 1/25) = 25/26`
`cos t = +- 5/sqrt26`
Since tan t > 0 and sin t > 0, therefor, cos t > 0
`cos t = 5/sqrt26`
`sin t = tan t.cos t = (1/5)(5/sqrt26) = 1/sqrt26`