Unit circle and y coordinate

Question

I was looking for some help with the y coordinate for the angle $\pi/6$. The coordinate is $\sqrt3/2$.

The x coordinate is easy to find.

The sin $\pi/6=.5$. .5 = 1/2. 1/2 can be found on the unit circle. The y coordinate is not as easy. The cos $\pi/6 = .8660254037784439$.

Is there an easy way to change the decimal form into a fraction? I tried the Pythagorean Theorem

.5 $^2$ + b $^2$ = 1$^2$

b= $\sqrt.75$

I felt the Pythagorean Theorem answer was closer to the unit circle but I am still having problems converting the number to the fraction $\sqrt3/2$. I was wondering if anyone would mind helping?

Thanks.

Answer 1

An equalateral triangle as angles each $\frac{\pi}{3}$. Draw a height to obtain two right triangles. Let the sides of the original equalateral triangle have length 2. So the hypotnuse of the right triangle has length 2, and the smaller leg has length 1. The smaller angle is $\frac{1}{2}\frac{\pi}{3}=\frac{\pi}{6}$ and by the Pythagorean theorem, the other leg has length $\sqrt{2^2-1^2}=\sqrt{3}$, thus $$ \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}.$$

Answer 2

You should convert the decimal inside the fraction and then separate the fraction in square root with fraction having square root in numerator and denominator.

$\sqrt{0.75}=\sqrt{\frac{75}{100}}=\sqrt{\frac{3}{4}}=\frac{\sqrt{3}}{\sqrt{4}}=\frac{\sqrt{3}}{2}$