# Reason why the differential calculus of sinθ becomes cosθ

The conclusion is described in Fig.1 .

The proof process will be described below.

sinθ and cosθ originally represent the relationship between two sides of the three sides of a triangle (Fig.2)

Fixing the length of the hypotenuse of the triangle to r and increasing the angle θ changes x and y to complete the circle. (Fig.3)

There are two types of units for the angle θ: [deg] (° or degrees) and [rad] (radians), but this time I will proceed with radians.

Some people may not be familiar with radians, but it is convenient to multiply the radius of a circle by radians to get the length of the arc (Fig.3).

In the first place, differential calculus is a little to move to see the change.

Let’s see what happens when I move the angle from θ radians by a little (dθ).

Since it is only a little (dθ), the circumference can be regarded as a straight line rather than a curve.

This time, set r = 1 for clarity.

One point to note before proceeding.

I saw the explanations drawn at K University and R University by like Fig.4 ,but it seems that it was drawn by bending the truth.

I have the impression that people who know only the answer without knowing the proof method is drawn to forcibly reach the answer.

If you think that you can understand by looking at the Fig.3 , be careful. It tends to be tricked by fake information.

Now, Fig.4 was drawn in equal proportion.

A space open to line segment BC is created

Some people may say, “Because they differentiate, line segment BC will be filled up.” It is people who are using the derivative as a sham and is not a proof as much as memorization.

Fig.6 is the one filled with line segment BC in Fig.5 .

The angles that are not shown will be shown in order.(Fig.7-10)

Since r = 1 this time, y = sin θ from ①.

The derivative of sinθ, that is, the derivative of y, dy / dθ, is approximated to cosθ from ⑥.

In other words, it can be said that the derivative of sin θ has almost the same value as cos θ.

sinθ、cosθとはそもそも三角形の３辺のうちのある２辺の関係を表すもの（図2）

ラジアンは使い慣れない人もいるだろうが、円の半径とラジアンを掛け合わせれば弧の長さが出るという便利なもの（図2）

またちょっと（dθ）だけなので円周は曲線ではなく直線とみなせる。

K大学、R大学が図4のような絵を描いて説明をしているのをみたが、

さて、等比率で描くと図5になる
これだと線分BCに開いた空間が出来てしまう

「微分するから線分BCは埋まるよ」という人が居るかもしれないが、
それは微分を誤魔化して使っている人で、暗記と大して変わらず証明になっていない。