# Oriented angles and polar locating : 11th grade math lesson

11th grade math lessons Report an error / Note? A 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.

Definition: trigonometric circle.

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. Ownership:

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 .

Definition:
Ownership:
The measurements of the angles in radian and degree are proportional.

## II. Measurements of an oriented angle.

Definition: 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 .

Property: main measure.
The angle oriented has a single measure in the interval called the principal measure of the angle.
Properties:
1. Chasles relation for angles: let , and be three non-zero vectors, then .
2. Characterization of the collinearity of two vectors: two vectors and are collinear if, and only if, .
Properties:

Let and be two non-zero vectors.

1. ;
2. 3. 4. ## III. Cosine and sine of a real and of an oriented angle.

### 1. Locating using cosine and sine.

Theorem : coordinates of a point of the trigonometric circle.

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

For any real x and any relative integer k.

1. (Pythagorean theorem).
2. 3. 4. 5. Particular values:  ### 2. The associated angles.

Properties:

For any real number x :

1. 2. 3. 4. 5. 6.  Properties:

For any real number x :

1. .
2. .
3. .
4. . ### 3. Duplication formulas.

Properties:

Consider two real numbers a and b.

1. .
2. .
3. .
4. .
Properties:

Consider a real number a.

1. .
2. .

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