# Homework Help: Exact value

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1. Jul 21, 2015

### 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.

2. Jul 21, 2015

### 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?

3. Jul 21, 2015

### Rectifier

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!