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**math lesson for 11th grade**on

**oriented angles and polar**. In this lesson, we will discuss sine, cosine and tangent with the different trigonometric relations.

## I. Locating on the trigonometric circle.

### 1. winding on the numerical right.

The **trigonometric circle** is the circle with center O and radius 1.

It has a direction of travel called **direct**, which is counter-clockwise.

With this choice, we say that the **plane is oriented**.

Any real number has a single image point on the circle If it exists such as then and have the same image point on the circle .

### 2. the radian

The radian measure of an angle is equal to the length of the arc of the trigonometric circle that it intercepts.

## II. Measurements of an oriented angle.

Let and be two non-zero vectors and the points M and N such that and are their respective representatives of origin O. Let M’ and N’ be the points of intersection of the half-lines [OM) and [ON) with the trigonometric circle.

Let x and y be two real numbers that have image points M’ and N’, then y-x is a radian measure of the oriented angle .

**principal measure**of the angle.

- Chasles relation for angles: let , and be three non-zero vectors, then .
- Characterization of the collinearity of two vectors: two vectors and are collinear if, and only if, .

Let and be two non-zero vectors.

- ;

## III. Cosine and sine of a real and of an oriented angle.

### 1. Locating using cosine and sine.

let x be a real number and M its image point on the trigonometric circle . The point M has coordinates .

For any real x and any relative integer k.

- (Pythagorean theorem).

Particular values:

### 2. The associated angles.

For any real number x :

For any real number x :

- .
- .
- .
- .

### 3. Duplication formulas.

Consider two real numbers a and b.

- .
- .
- .
- .

Consider a real number a.

- .
- .

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