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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! 🙂