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

Hi.
By definition ##tantheta=sintheta/costheta## with ##theta## being the angle.
But be careful that makes us wish to say that:
##tantheta=sintheta/costheta=1/2=>sintheta=1 and costheta=2## and that is not correct! Cosine and Sine are functions that oscilate between ##-1## and ##1## never (never? Never!) above or below.
Now let’s explore the definition
##sintheta/costheta=1/2=>2sintheta=costheta##
Squaring both sides of the last equality
##2^2sin^2theta=cos^2theta##
We’re going to use the trig identity: ##sin^2theta+cos^2theta=1## but first let’s separate it
##sin^2theta+cos^2theta=1=>cos^2=1-sin^2theta## and put this in our equality gives us:
##2^2sin^2theta=1-sin^2theta##
##4sin^2theta+sin^2theta=1##
##5sin^2theta=1=>sin^2theta=1/5=>sintheta=1/sqrt5##
Simplifying
##sintheta=sqrt5/sqrt5(1/sqrt5)=sqrt5/5##
All we need to do next is to use the first definition and it’s over:
##tantheta=sintheta/costheta=(sqrt5/5)/costheta=1/2## cross products!
##costheta=2sqrt5/5##
There are other ways of doing this feel free to try!
Hope it helps! 🙂