# Pi Day Countdown #9 – Pi and the triangle Part 1

The study of the triangles or trigonometry is an ancient branch of geometry studied by the Greeks, Indians and Babylonians. Trigonometry deals with the study of angles and lengths of a triangle, and indirectly the circle, because of this, trigonometry gives a powerful tool to approximate and solve Pi.

One of the ways that trigonometry can be used to approximate Pi is via equations known as trig formulas. One such formula is called arctangent ($arctan(x)$) which as a function, will give the degree of any angle, based off of the ratio of the opposite and adjacent sides. $arctan(x)$ however will only work for right triangles and the reference angle cannot be the right angle.

Now since it takes a 360 degree rotation to form a circle and since the circumference of the unit circle is $2 \pi$, this means that degrees can also be represented by fractions of $2 \pi$. This unit if measure is called a radian. For example $60^{\circ}$  is $\pi / 3$, $45^{\circ}$ is $\pi / 4$.

So with proper values, an approximation of $arctan(x)$ will give an approximation of $\pi$. A good case to start with is $arctan(1)$. This right triangle has an angle of degree $45^{\circ}$, which means in radians the angle is worth $\pi / 4$.

Eureka! All we need now is to find an approximation for $arctan(1)$ and we will be able to find an approximation for $\pi / 4$.

The approximation technique will be continued in part 2.

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