math
9th Grade - Trigonometry

9th Grade lesson

5 - Trigonometry


Take two perpendicular and graduated lines. Then draw a circle of radius 1 and whose center is the intersection of the lines.

cercle dans repère

Now, draw a line passing through O ; the angle between this line and the horizontal axis is 40 °. Denote by M the intersection of this line and the circle. The abscissa of M is cos(40) and its ordinate is sin(40).

circle

The angle you chose doesn't matter : M is on the circle, you'll always obtain :

property du cosine
property du sine

When you move M to the right, its cosine gets closer to 1 and its sine gets closer to 0 ; when it goes up, its cosine gets closer to zero and its sine gets closer to 1. Sine et Cosine are functions : give them angles (degrees or radians), they will give you numbers between -1 and 1 in return.

The radian

The radian is a unit of angular measurement. If the length of the red arc is x, then the angle measures x radians.
definition du radian

The whole circle measures 360 degrees. In radians, it measures the circumference of the circle, that is 2number pi (the radius of the circle is still 1). In the drawing, the angle x is approximately 1 radian. As you know that 2number pi rad=360°, you can convert, thanks to a cross-product, degrees into radians and radians into degrees. Hence :

conversion radian degre

Table of values : sine and cosine

You have to know by heart the cosine and sine values of the angles below (to learn them, you can learn the drawing).

valeurs du sine et du cosine
angle x 0
angle in radius
angle in radius
angle in radius
angle in radius
angle in radius
cos x
1
cosine angle
cosine angle
cosine angle
0
- 1
sin x 0
sine de angle
sine de angle
sine de angle
1
0


Thanks to Pythagoras' theorem, in the triangle ONM which is right-angled at N, we have : pythagorean theorem in a circle, and since the radius of the circle is 1, we have : link between sine and cosine circle

Graph

If angles are greater than 2number pi, the point M turns around the circle ; when it is 2 number pi radians, it comes back to the same place. Hence :
cosine function
sine fonction periodique

For example cosine calculation, so you can draw, thanks to the table of values, the graph of the cosine function, and the graph of the sine function :

sine graph

You can prolong the two graphs infinitely. On the graph, we can see than cosine is even (symmetric with respect to the Y-axis ), and the sine is odd (symmetric with respect to the origin).

Tangent

We won't say much about the tangent function on this page. Nevertheless, you have to know that :
tangent formula



See also : trigonometry on fmaths.com



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9th Grade Trigonometry

lesson, problems