Calculate the Unknown Angle of a Right Triangle

[Rectifier]

The problem

A right triangle has an angle a and we know that $cos \ a = \frac{1}{3}$. What is $tan \ (90°-a)$

The attempt
I know that the ration between the adjacent side and the hypothenuse is 1/3. I am not interested in the real lengths of the sides.

I can therefore calculate the possible length of the opposite side of the angle a by applying the Pythagorean theorem.

$$ 1^2+x^2=3^2 \\ 1+x^2=9 \\ x = \sqrt{8}= 2 \sqrt{2} $$

I can now write out tan(a):

$$ \tan(a)=\frac{\sqrt{8}}{1}=\sqrt{8} = 2 \sqrt{2}$$

And here is where I get stuck :,(

Note:
I have to calculate the angle without any calculator.

 

[DEvens]

Is there an angle in your problem that is equal to 90 - a? If you figure that out, then can you find the tan of that angle?

 

[Rectifier]

Is there an angle in your problem that is equal to 90 - a? If you figure that out, then can you find the tan of that angle?

Oh!
sin(90-a)=cos(a) and cos(90-a)=sin(a)! :D

this means that

$$ tan (90-a)= \frac{sin(90-a)}{cos(90-a)}= \frac{cos(a)}{sin(a)}=cot(a)$$

Thank you so much!