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

2 Answers
Aug 4, 2018

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

Aug 4, 2018

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