Oriented angles and polar locating : 11th grade math lesson

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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 \varphi 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.

trigonometric circle


Any real number x has a single image point on the circle \varphiIf it exists k\in\,\mathbb{Z} such as x'=x+2k\pithen x and x' have the same image point on the circle \varphi.

2. the radian


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


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

II. Measurements of an oriented angle.

Definition: oriented angle.

Let \vec{u} and \vec{v} be two non-zero vectors and the points M and N such that \vec{OM} and \vec{ON} 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 (\vec{u};\vec{v}).

Property: main measure.
The angle oriented (\vec{u};\vec{v}) has a single measure \alpha in the interval %5D-\pi:\pi%5B called the principal measure of the angle.
  1. Chasles relation for angles: let \vec{u},\vec{v} and \vec{w} be three non-zero vectors, then (\vec{u},\vec{v})+(\vec{v},\vec{w})=(\vec{u},\vec{w}).
  2. Characterization of the collinearity of two vectors: two vectors \vec{u} and \vec{v} are collinear if, and only if, (\vec{u},\vec{v})=0%5B\pi%5D.

Let \vec{u} and \vec{v} be two non-zero vectors.

  1. (\vec{v},\vec{u})=-(\vec{u},\vec{v});
  2. (-\vec{u},-\vec{v})=\,(\vec{u},\vec{v})
  3. (-\vec{u},\vec{v})=\,(\vec{u},\vec{v})%5B\pi%5D
  4. (\vec{u},-\vec{v})=\,(\vec{u},\vec{v})%5B\pi%5D

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 \varphi. The point M has coordinates (cosx;sinx).

cosx sinx


For any real x and any relative integer k.

  1. cos^2x+sin^2x=1 (Pythagorean theorem).
  2. -1\leq\,\,cosx\leq\,\,1
  3. -1\leq\,\,sinx\leq\,\,1
  4. cos(x+2k\pi)=cosx
  5. sin(x+2k\pi)=sinx

Particular values:

table values cos sin

trigo circle

2. The associated angles.


For any real number x :

  1. cos(-x)=cosx
  2. sin(-x)=-sinx
  3. cos(\pi-x)=-cosx
  4. sin(\pi-x)=sinx
  5. cos(\pi+x)=-cosx
  6. sin(\pi+x)=-sinx



For any real number x :

  1. cos(\frac{\pi}{2}-x)=sinx.
  2. sin(\frac{\pi}{2}-x)=cosx.
  3. cos(\frac{\pi}{2}+x)=-sinx.
  4. sin(\frac{\pi}{2}+x)=cosx.


3. Duplication formulas.


Consider two real numbers a and b.

  1. cos(a+b)=cosa\times  \,cosb-sina\times  \,sinb.
  2. cos(a-b)=cosa\times  \,cosb+sina\times  \,sinb.
  3. sin(a+b)=sina\times  \,cosb+cosa\times  \,sinb.
  4. sin(a-b)=sina\times  \,cosb-cosa\times  \,sinb.

Consider a real number a.

  1. cos(2a)=cos^2\,a-sin^2\,a\\=2cos^2\,a-1\\=1-2sin^2\,a.
  2. sin2a=2\times  \,cosa\times  \,sina.

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