### Gauss' trigonometric identities for heptadecagon

Today, we write down Gauss' magical trigonometric identities for regular heptadecagon.

### Construction algorithm

In our previous post, we have learnt about

*Similar Triangles*and the

*Right Triangle Altitude Theorem*. Today, continuing our journey in the garden of geometry, we want to find answer to the following question

Given a line segment of length $r$, by compass and straightedge, what kind of shapes can we construct?

### Right Triangle Altitude Theorem

*Right Triangle Altitude Theorem*and use it to derive

*Pythagorean Theorem*.

### How to construct a regular polygon with 15 sides

To feed your curiosity, today we will look at a compass and straightedge construction of the regular polygon with 15 sides and we will show that there is a connection between this construction problem with the

*measuring liquid puzzle*that we have learned from our previous post.

### Measuring liquid puzzle

Today we will look at a brainteaser puzzle: "

*how to measure out exactly*." We will analyse this puzzle to see that, despite its innocent look, this puzzle has a close connection with the linear Diophantine equation.

**1 liter**of water using a**3-liter**jug and a**5-liter**jug### Construction of regular polygons

**constructing regular polygons**and the problem of

**trisecting an angle**by compass and straightedge. It was not until the 18th-19th centuries that mathematicians could finally solve them by employing advanced tools of number theory and algebra. Today, we will look at the regular polygon construction problem.

Regular polygon construction problem.Using compass and straightedge, construct a regular polygon with $n$ sides.

### Trigonometric multiple-angle formulas

We derive this formula of $\cos{\frac{\pi}{5}}$ by observing that $\cos{\frac{2 \pi}{5}} = -\cos{\frac{3 \pi}{5}}$ and then applying the trigonometric formulas for double angle and triple angle:

$$\cos{2 x} = 2 \cos^2{x} - 1,$$ $$\cos{3 x} = 4 \cos^3{x} - 3 \cos{x}$$ to set up a cubic equation for $\cos{\frac{\pi}{5}}$.

It seems a good occasion now for us to learn about trigonometric multiple-angle formulas. In this post, we will show how to derive formulas for $\sin{nx}$, $\cos{nx}$, $\tan{nx}$ and $\cot{nx}$ using de Moivre's identity of the complex numbers.

### Construction of a regular pentagon

*regular pentagon*based on the following trigonometric formula $$\cos{\frac{\pi}{5}} = \frac{1 + \sqrt{5}}{4}.$$

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