Question

How do you find the values of `sin 2theta` and `cos 2theta` when `cos theta = 12/13`?

Answer 1

Below

Explanation:

`theta` can be in the first quadrant `0<=theta<=90` or the fourth quadrant `270<=theta<=360`

If `theta` is in the first quadrant,
then
`sintheta=5/13`
`costheta=12/13`
`tantheta=5/12`

Therefore,
`sin2theta=2sinthetacostheta=2times5/13times12/13=120/169`

`cos2theta=cos^2theta-sin^2theta=(12/13)^2-(5/13)^2=144/169-25/169=119/169`

If `theta` is in the fourth quadrant,
then
`sintheta=-5/13`
`costheta=12/13`
`tantheta=-5/12`

Therefore,
`sin2theta=2sinthetacostheta=2times-5/13times12/13=-120/169`

`cos2theta=cos^2theta-sin^2theta=(12/13)^2-(-5/13)^2=144/169-25/169=119/169`

Answer 2

`sin 2theta = 120/169, theta in Q_1 and - 120/169, theta in Q_4`.

`cos 2theta = 119/169`

Explanation:

See my answer in

https://socratic.org/questions/if-cos-a-5-13-how-do-you-find-sina-and-tana

As a continuation,

`sin theta = 5/13, theta in Q_1` and it is `- 5/13, theta in Q_4`.

`sin 2theta = 2 sin theta cos theta = 2 ( 5/13)(12/13)`

`= 120/169, theta in Q_1 and`

`= 2 ( -5/13)(12/13) = - 120/169, theta in Q_4`.

`cos 2theta = cos^2theta - sin^2theta = (12/13)^2 - (+-5/13)^2`

`= 119/169`