Trigonometry: 2nd grade math course to download in PDF.

Indeed, it should be known that trigonometry is a very important chapter and requires the use of certain geometric materials.

A 10th grade math course on trigonometry and more specifically on the sine and cosine functions will be beneficial to you. We will discuss the trigonometric circle and the periodicity of these functions as well as their respective representative curves. Later, we will discover the trigonometry formulas involving cosine and sine.

I. The trigonometric functions

In this lesson, (O,\vec{u},\vec{v}) is an orthonormal coordinate system with a direct direction.

The points A and B are thus on the trigonometric circle of center O and radius 1.

trigonometric circle

1.definition of the sine and cosine of a real number.


To any real \alpha, we associate the point M of the trigonometric circle such that the oriented angle (\vec{u},\vec{OM}) measures \alpha radian(s).

The cosine and sine of \alpha are therefore the coordinates of M in the reference frame (O,\vec{u},\vec{v}).

We have: M(cos\alpha\,,sin\,\alpha\,) that is to say: \vec{OM}=cos\alpha\,\vec{u}+sin\alpha\,\vec{\,v}.

oriented angle

2. first properties in trigonometry .

  • If \alpha=0 then the point of the trigonometric circle associated to \alpha is the point A(1; 0). So cos(0) = 1 and sin(0) = 0
  • If \alpha\,=\frac{\pi}{2}, then the point of the trigonometric circle associated with \alpha is B(0; 1).Therefore cos(\frac{\pi}{2})=0 and sin(\frac{\pi}{2})=1.
  • If \alpha\,=\pi, then x is associated with A'(-1 ;0). So cos(\,\pi\,)=-1 and sin(,\pi,)=0.
  • If \alpha\,=-\frac{\pi}{2} then \alpha is associated with B'(0 ;-1). So cos(-\frac{\pi}{2})=0 and sin(-\frac{\pi}{2})=-1.
  • If \alpha is a real then for any relative integer k, the reals \alpha and \alpha\,+2k\pi are associated with the same point M.
    Indeed, these are two measures of the angle oriented.
    So, for any real number x and any relative integer k, we have



We say that the cosine and sine functions are periodic with period 2\pi because T = 2\pi is the smallest strictly positive real such that: cos (\alpha + T) = cos \alpha and sin (\alpha + T) = sin \alpha .

The Pythagorean theorem allows to prove the equality:
(sin\,\alpha\,)^2\,+\,(cos\,\alpha)^2\,=\,1 which is also written as: sin^2\,\alpha\,+\,cos^2\,\alpha\,=\,1.

3. sign of sine and cosine in trigonometry

By definition, the sine and cosine of any real number belong to the interval [-1; 1].

Specifically, the position of M tells us more about the cosine and sine of \alpha.


We have:

  • If \alpha\,\in%5B0+2k\pi,\pi+\,2k\pi%5D then sin\alpha\,\geq\,\,0.
  • If \alpha\,\in%5B-\frac{\pi}{2}+2k\pi\,\frac{\pi}{2}+\,2k\pi%5D then cos\alpha\,\geq\,\,0.

II. Cosines and sines of remarkable angles in trigonometry

All these results to know perfectly are summarized in the table below:

table cos sin

III. Visualization of sine and cosine on the trigonometric circle.

This is an indispensable tool, which it is useful to visualize well in order to be able to quickly find the values indicated below.

cos sin

IV. Usual formulas concerning the associated angles.


For any real x, we have:

cos(-x)\,=\,cos(x) and sin(-x)\,=\,-sin(x).

The cosine function is therefore even and the sine function is odd.


For any real x, we have:

cos(\pi – x) = – cos(x) and sin(\pi – x) = sin(x).


For any real x, we have:

cos(\pi + x) = – cos(x) and sin(\pi + x) = – sin(x).


For any real x, we have:

cos(\frac{\pi}{2}+x) = – sin(x) and sin(\frac{\pi}{2}+x) = cos(x).


For any real x, we have:

cos(\frac{\pi}{2}-x) = sin(x) and sin(\frac{\pi}{2}-x) = cos(x).

V. Graphical representations of sine and cosine functions in trigonometry

sine cosine curves

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