What is sin and cos if# tan = 1/2# and #sin >0#?

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Answer 1 · 2015-04-14T14:15:08

The answers are:

#sinalpha=sqrt5/5#,

#cosalpha=2sqrt5/5#.

First of all, if the sinus and the tangent of an angle #alpha# is positive, the angle is in the first quadrant and so sinus and cosine are positive!

Since #tanalpha=sinalpha/cosalpha#,

than:

#sinalpha/cosalpha=1/2#

and, for the fundamental relation of trigonometry:

#sin^2alpha+cos^2alpha=1#,

let's solve the system!

#cosalpha=2sinalpha#

#sin^2alpha+cos^2alpha=1#

than

#sin^2alpha+4sin^2alpha=1rArrsin^2alpha=1/5rArrsinalpha=sqrt5/5#

(only the positive value!)

and

#cosalpha=2sinalpha=2sqrt5/5#.