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! :)