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

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

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