1. irrational numbers

prove that the following are irrational numbers

1) tan 5

2) log 6 to the base 7

2. Hello, fxs12!

Here's the second one . . .

$\displaystyle \text{Prove that the following are irrational numbers: }$

. . $\displaystyle (1)\;\tan 5 \qquad\qquad (2)\;\log_76$

Suppose $\displaystyle \log_76$ is rational.

Then: .$\displaystyle \log_76 \:=\:\dfrac{n}{d}\:\text{ for some integers }n,d$

. . And we have: .$\displaystyle 7^{\frac{n}{d}} \:=\:6$

Raise both sides to the power $\displaystyle \,d\!:\;\;7^n \:=\:6^d$

. . And we have: .$\displaystyle 7^n \;=\;2^d\cdot3^d$

We have a number that has two distinct prime factorizations.
This is contradiction of the Fundamental Theorem of Arithmetic.

Therefore: .$\displaystyle \log_76$ is irrational.

3. Originally Posted by fxs12
prove that the following are irrational numbers

1) tan 5

[snip]
Since $\displaystyle \tan (n \theta^{0})$ is a rational function of $\displaystyle \tan (\theta)$ it can be deduced from the fact that $\displaystyle \tan 60^{0} = \sqrt{3}$ that $\displaystyle \tan (n^{0})$ is irrational for n = 1, 2, 3, 4, 5, 6, 10, 12, 15 or 30.

4. Is this $\displaystyle \displaystyle \tan{(5^{\circ})}$ or $\displaystyle \displaystyle \tan{(5^C)}$?