Trigonometric function

The trigonometric functions are

Where the variable α is the radian measure of an angle. The geometrical meaning of the four trigonometric functions can be illustrated with the unit circle: a circle that has its center at the origin of coordinates and its radius is. We can see that the sine, cosine and tangent and cotagent of the angle can be found using the following relations:

Since the radius of the unit circle has length r = +1, the sine and cosine are given by the ordinate and abscissa, respectively (with the appropriate sign). tg α = lenght with sign of the segment BK, cotg α = lengt with sign of the segment PL. The following fundamental relations hold true

The first relations follows from the Pythagorean theorem. Fot the second notice that in the Figure above the value of the tangent function is displayed by the vertical segment line BK passing through the point B(1,0). Indeed, the triangles OAM and OBK are similar. Therefore,

Here AM = OS = y = sin α, OA = x = cos α, OB = 1. Hence

$$BK=\frac{AM}{OA}\times OB=\frac{\sin \alpha }{\cos \alpha }\times 1=\tan \alpha .$$

In the Figure above, the cotangent of α is numerically equal to the length of the horizontal segment line PL. Since the triangles OSM and OPL are similar, we have

Then

$$PL=\frac{SM}{OS}\times OP=\frac{\cos \alpha }{\sin \alpha }\times 1=\cot \alpha .$$

1.4 Definition: A function f is said to be periodic if, for some nonzero constant T, it is the case that

The principal properties of the trigonemetric functions are listed in the following table.

	x ⟼ sin x	x ⟼ cos x	x ⟼ tg x	x ⟼ cotg x
Domain	ℝ	ℝ	ℝ \ {π/2 + kπ}	ℝ \ {kπ}
Range	[−1,1]	[−1,1]	ℝ	ℝ
Continuity interval	ℝ	ℝ	continuity in the domain	continuity in the domain
Period	2π	2π	π	π
Parity	sin (−x) = −sin x Odd function	cos (−x) = cos x Even function	tg (−x) = −tg x Odd function	cotg (−x) = −cotg x Odd function

In particular if x₀ ∈ ℝ we have that

$$\underset{x\to x_0}{lim}\sin x=\sin x_0$$ $$\underset{x\to x_0}{lim}\cos x=\cos x_0$$ $$\underset{x\to x_0}{lim}\tan x=\tan x_0\text{if }{x}_0\ne \frac{\pi }{2}+k\pi ,k\in \mathbb{Z}$$ $$\underset{x\to x_0}{lim}\cot x=\cot x_0\text{if }{x}_0\ne k\pi ,k\in \mathbb{Z}$$

Remark. Note that for all four functions the limits lim_{x ⟶ ∞} f(x) do not exist. This is true in general for all periodic functions.

Since sin x = cos (x − π/2), we obtain that sin x is phase-shifted by π/2, and the graph of sin x is obtained shifting the graph of sin x to the right by π2.

Notice that for both the sine and cosine functions the domain is -∞, ∞ and the range is the closed interval [−1, 1]. Thus, for all values of x, we have:

−1 ≤ sin x ≤ 1;

−1 ≤ cos x ≤ 1;

o in termini di valore assoluto:

|sin x| ≤ 1;

|cos x| ≤ 1;

Also, the zeros of the sine function occur at the integer multiples of π that is,

sin x = 0 when x = nπ with n integer

Reciprocal trigonometric functions

The reciprocal of sine is cosecant: csc θ = 1/sin θ

The reciprocal of cosine is secant: sec θ = 1/cos θ

The reciprocal of tangent is cotangent: cot θ = 1/tan θ

The graph of the cotangent function is similar to the graph of the tangent function. It also has a period of π. However, from the identity

you can see that the cotangent function has vertical asymptotes when sin x is zero.

to sketch the graph of a secant or cosecant function, I'll lightly draw the sine wave, and draw vertical asymptotes through its zeroes and note the min/max points

and the cosecant function

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