First of all, we should know that trigonometric function is defined by unit circle. The unit circle is a circle with a radius of 1, and its center is at the origin. On the unit circle, we can choose any point as the starting point, and then draw an angle intersecting the unit circle with that point as the vertex. The size of this angle is the value of the trigonometric function we want to calculate.
For example, the sine function sin is defined as the ratio of the length of the opposite side to the length of the hypotenuse of an angle on the unit circle. Suppose we choose a starting point a on the unit circle, and then draw an angle BAC that intersects the unit circle. Then, the value of sine function sin is the ratio of the length of the opposite side BC of the angle BAC to the length of the hypotenuse AC.
Similarly, cosine function cos and tangent function tan are defined in a similar way. The value of cosine function cos is the ratio of the length of adjacent side AB of angle BAC to the length of hypotenuse AC, while the value of tangent function tan is the ratio of the length of opposite side BC of angle BAC to the length of adjacent side AB.
These definitions are understood in a geometric intuitive way, but if we want to prove the correctness of these definitions in a mathematical way, we need to use advanced mathematics knowledge. This requires us to use some basic knowledge of calculus and linear algebra, such as limit, derivative, matrix and so on.
For example, we can use derivatives to prove the periodicity of sine and cosine functions. We know that if a function satisfies f(x+2π)=f(x), it is called a periodic function, where 2π is the period of this function. We can use derivative to prove that sine function and cosine function satisfy this condition.
First, we can calculate the derivatives of sine function and cosine function. For sine function sin, its derivative is cos；; For cosine function cos, its derivative is -sin. Then we can calculate the periodicity of these two derivatives. Through calculation, we can find that cos(x+2π)=cos(x), -sin(x+2π)=-sin(x). This proves that sine function and cosine function are both periodic functions, and their periods are both 2π.
In this way, the calculation formula of trigonometric function can be deduced by means of advanced mathematics. These formulas are not only of theoretical significance, but also very important in practical application.