Fourier Transform from Fourier Series

Before you see this post,

see

Definite Integral of Multiplied 2 Trigonometric Functions

Taylor Series

Express Functions from Taylor Series

We can think f(t) is sum of wave functions and a constant.

if t=0, f(0) = a_0.

Then, let's find a_0.

We can find a_0 like process above.

Through

Definite Integral of Multiplied 2 Trigonometric Functions,

we can find a_n ( except n = 0 ) and b_n.

Use

Express Functions from Taylor Series,

we can develop the formula.

In the process, we can derive Fourier Transform from Fourier Series.

Differential of A Function : sin x, cos x

Before you see this post,

see

A Property of Limit of A Function : f(x)/g(x)

A Property of Limit of A Function : f(x)g(x)

Synthesis of Trigonometric Functions

Synthesis of Trigonometric Functions

Before you see this post,

see

Distance between A Point and A line

and

Sum and Products by Unit Vector

Limit of sin(x)/x and tan(x)/x

Before you see this post,

see

A Property of Limit of A Function : f(x)/g(x),

Trigonometric Function : Sine, Cosine and Tangent,

Graph of Trigonometric Function : Sine, Cosine and Tangent

and

Radian

First, Let's divide the formula by sinθ.

There is a result.

Second, Let's divide the formula by tanθ. 

There is a result.

Graph of Trigonometric Function : Sine, Cosine and Tangent

Before you see this post,

see

Trigonometric Function : Sine, Cosine and Tangent

Trigonometric Function : Sine, Cosine and Tangent

               In Right-angled triangle, they are defined like the picture below.

sine -> sin

cosine -> cos

tangent -> tan

In Cartesian coordinate system, 

we can notate trigonometric functions, sine, cosine and tangent, 

like the picture above.