Definition of Cos

Cos is one of the trigonometric ratios. The six basic trigonometric ratios are sin, cos, tan, sec, cosec and cot. Of these sin and cos are most familiarly used. Trigonometric ratios involve angles in their standard positions. They can be defined using a Right triangle as well as by a Circle.

Let us see the Definitions of Cos

Definition 1: using a right triangle



The sides of a right triangle are

1. Hypotenuse : The side opposite to the right angle (`90^o` ), in this case side `BC`
2. Opposite side: The side opposite to the angle of consideration (angle `theta` ), in this case side `AB`
3. Adjacent side: The side adjacent to the angle of consideration(angle `theta` ), here it is side `AC`

Definition of cos : Cosine is defined as the ratio of adjacent side to the hypotenuse

So, `cos(theta) = (adjacent side) / (hypoten use)`

`= (AC)/(BC)`

Here cos is defined for angles  `0^o< theta < 90^o`

Note: here ratio does not depend on the lengths of a particular triangle, it is same for all the right triangles which contain the  angle `theta` . Because all such right triangles will be simillar.

Definition 2: Using  Circle

Consider a circle with center  ‘`O` ’ at origin and radius ‘`r` ’ in the rectangular cartesian system. Let `theta` be any angle in standard position such that its terminal ray intersects the circle in point P(x,y).



From the figure

We have `OP = r`

And `x^2+y^2 = r^2`

Now cosine is defined as   `cos(theta)=x/r`

Here cos is defined for angles `0^o < theta < 360^o`

Examples for Definition of Cos :

For  a triangle  with sides AB =3, BC=5 ,AC=4 Right angles at ‘`A` ’


Calculate  `CosB` and `CosC`

Solution:

For calculating `cosB` :



Adjacent side is AB =3 , opposite side is AC =4, Hypotenuse is BC= 5

So `cos B = (adjacent side)/( hypoten use)`

`=(AB)/(BC)`

` = 3/5`

For calculating `Cos C` :



Adjacent side is AC =4, Opposite side is AB =3, Hypotenuse BC =5

So, `cos C = (adjacent side)/ (hypoten use)`

` = (AC)/( BC)`

` = 4/5`