| Rational Trig Functions |
Demonstrate that if the tangent of half of an angle is rational, any trigonometric function of the whole angle will (if defined) be rational.
Submitted by Sasha |
Problem Moderated by: Douglas |
| Problem Solution |
The relevant trig identities for half angles are as follows:
sin(x) = 2 tan(x/2)/(1+tan2(x/2))
cos(x) = (1-tan2(x/2))/(1+tan2(x/2)
Also, quotients formed from rational numerators and denominators are also rational.
Since we are given that tan(x/2) is rational, the numerators and denominators of the above relations are rational, and hence the right-hand sides are rational. Therefore sin(x) and cos(x) are rational. Because tan(x)= sin(x)/cos(x), sec(x)=1/cos(x) and cosec(x)=1/sin(x), it follows that the six common trig functions are all rational.
Solution submitted by Donnie |
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