3,033

edits
# Changes

Created page with "{{particular function}} {{angular function radian convention}} ==Definition== ===Unit circle definition=== The '''cosine function''', denoted <math>\cos</math>, is defined as f..."

{{particular function}}

{{angular function radian convention}}

==Definition==

===Unit circle definition===

The '''cosine function''', denoted <math>\cos</math>, is defined as follows.

Consider the unit circle centered at the origin, described as the following subset of the coordinate:

<math>\{ (x,y) \mid x^2 + y^2 = 1\}</math>

For a real number <math>t</math>, we define <math>\cos t</math> as follows:

* Start at the point <math>(1,0)</math>, which lies on the unit circle centered at the origin.

* Move a distance of <math>t</math> along the unit circle in the counter-clockwise direction (i.e., the motion begins in the first quadrant, with both coordinates positive).

* At the end, the <math>x</math>-coordinate of the point thus obtained is defined as <math>\cos t</math>.

===Triangle ratio definition (works for acute angles)===

For an acute angle <math>t</math>, i.e., for <math>t</math> in the [[open interval]] <math>(0,\pi/2)</math>, <math>\cos t</math> can be defined as follows:

* Construct any right triangle with one of the acute angles equal to <math>t</math>.

* <math>\! \cos t</math> is the ratio of the leg adjacent to the angle <math>t</math> to the hypotenuse.

{{angular function radian convention}}

==Definition==

===Unit circle definition===

The '''cosine function''', denoted <math>\cos</math>, is defined as follows.

Consider the unit circle centered at the origin, described as the following subset of the coordinate:

<math>\{ (x,y) \mid x^2 + y^2 = 1\}</math>

For a real number <math>t</math>, we define <math>\cos t</math> as follows:

* Start at the point <math>(1,0)</math>, which lies on the unit circle centered at the origin.

* Move a distance of <math>t</math> along the unit circle in the counter-clockwise direction (i.e., the motion begins in the first quadrant, with both coordinates positive).

* At the end, the <math>x</math>-coordinate of the point thus obtained is defined as <math>\cos t</math>.

===Triangle ratio definition (works for acute angles)===

For an acute angle <math>t</math>, i.e., for <math>t</math> in the [[open interval]] <math>(0,\pi/2)</math>, <math>\cos t</math> can be defined as follows:

* Construct any right triangle with one of the acute angles equal to <math>t</math>.

* <math>\! \cos t</math> is the ratio of the leg adjacent to the angle <math>t</math> to the hypotenuse.