Why is the unit circle useful

Cosine and sine on the unit circle

We define:

definition(Circle)

Let r ∈ ℝ+0. Then is called

Kr = {(x, y) ∈ ℝ2 | x2 + y2 = r2 }

the circle or more precisely the Circle line With radius r and Focus 0 = (0, 0) of level ℝ2. The circle K = K1 is called that Unit circle in ℝ2.

definition(Angle in radians)

If P is a point on the unit circle K, then the length α ∈ [0, 2π [of the circular arc of K, which leads counterclockwise from (1, 0) to P, that of P and the x-axis included angles (in radians)in the interval [0, 2π [.

We agree:

Convention

By identifying real numbers that differ only by an integral multiple of 2π, we can interpret every real number as an angle. For all α ∈ ℝ there is then a point P on the unit circle with the angle α. Two angles that correspond to the same point P differ by an integral multiple of 2π.

This identification clearly corresponds to running through the unit circle multiple times for α ≥ 2π or running through the unit circle clockwise for α <0. It is often useful to choose angles in the interval] −π, π] instead of [0, 2π [ . In principle, every half-open interval of length 2π is suitable.

After these preparations we can now introduce the cosine and sine functions.

definition(Cosine and sine as coordinate functions)

Let α ∈ ℝ, and let P be the point of the unit circle K with angle α. Then we set

cos (α) = "The x-coordinate of P",
sin (α) = "The y-coordinate of P".

The functions cos: ℝ → ℝ and sin: ℝ → ℝ defined in this way are called the Cosine function or. Sine function on ℝ.

So if P lies on K with an angle α, then by definition

P = (cos (α), sin (α)).

Because of our identification of angles, the cosine and sine functions are 2π-periodic; H. it applies

cos (α + k2π) = cos (α),

sin (α + k2π) = sin (α) for all α ∈ ℝ and k ∈ ℤ.

We can look at the definition from different angles. Two important interpretations are:

Dynamic interpretation: even circular motion

If a point P moves uniformly counterclockwise on the unit circle at the time t with the angular velocity 1 and the starting point P (0) = (1, 0), then the following applies

P (t) = (cos (t), sin (t)) for all t ∈ ℝ.

Geometric interpretation: true-to-length circular winding

Let x ∈ ℝ. If we wind a segment from 0 to x true to length on the unit circle (counterclockwise for x ≥ 0, clockwise for x <0), this circular winding ends at the point

P (x) = (cos (x), sin (x)) ∈ K.

When dealing with trigonometric functions, we use the variables x, y, α, β, ..., φ, ψ, ... depending on the context and in principle completely freely. In order to save brackets, we agree:

notation

We often leave out function brackets and write sin x, cos x instead of sin (x), cos (x). We continue to write sin2 x, cos2 x instead of (sin x)2, (cos x)2.

In general, we also use the alternative notation fx for each function f in addition to f (x), wherever it is for the sake of clarity.