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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 .
- 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).
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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