can someone explain this to me:
tan(theta) = 60/x
x = 60/tan(theta)
x = 60cot(theta)
i dont understand how the tangent switches to cotangent.
Cotangent is just 1 over tangent so
$\displaystyle cot(x) = \frac 1{tan(x)}$
In the same way that cosecant is 1 over sine, and secant is 1 over cosine.
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So your intitial equation is:
$\displaystyle tan(\theta)=\frac{60}x$
then multiply both sides by $\displaystyle \frac{x}{tan(\theta)}$
$\displaystyle \frac{x}{tan(\theta)}*tan(\theta)=\frac{60}x*\frac {x}{tan(\theta)}$
Simplify (cancel out tangents on LHS and x's on RHS) to get your second equation.
$\displaystyle x=\frac{60}{tan(\theta)}$
You can partition the RHS like so:
$\displaystyle x=60*\frac{1}{tan(\theta)}$
And you see that on the RHS 1/tangent = cotangent so
$\displaystyle x=60*cot(\theta)$
Which is your last equation.
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Some other things to note:
since $\displaystyle cot(\theta) = \frac 1{tan(\theta)}$ you can multiply both sides by tangent over cotangent and get $\displaystyle tan(\theta)=\frac 1{cot(\theta)}$
and also since $\displaystyle tan(\theta) = \frac{sin(\theta)}{cos(\theta)}$
it follows that $\displaystyle cot(\theta) = \frac 1{tan(\theta)} = \frac{1}{\frac{sin(\theta)}{cos(\theta)}} = \frac{cos(\theta)}{sin(\theta)}$
so $\displaystyle cot(\theta)=\frac{cos(\theta)}{sin(\theta)}$