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Thought-Process to Discover Knowledge

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mathsTrigonometryBasics of Trigonometry

Trigonometric Ratios Explained (revolutionary)

Get ready for the revolutionary and awe-inspiring explanation to trigonometric ratios. This is the only place in the world to provide such insights and revelations.



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We have learned

 •  Many applications are mathematically modeled using the sides of right angled triangles.

 •  the set of similar right angled triangles are defined by an angle.

Though the set of similar right angled triangles is specified by an angle, to narrow down to a specific right angled triangle, one of the sides is specified.

A right angled triangle is specified by

 •  an angle - that specifies the set of similar right angled triangles

 •  length of one of the three sides of the triangle

When using this, it may be required to find the length of other sides of the triangle as the specified side may not be the one required in the calculations.

Angle defines set of similar right angled triangles. Providing one of the sides narrows it down to a specific right angled triangle.

Specifying Right Angled Triangle: Set of similar right angled triangles are defined by one parameter: "angle". And to narrow down to one specific triangle, one side is additionally specified.

trigonometric ratios of similar triangle In a right angled triangle, When an angle and one of the sides is specified, what does one require to compute the other sides of the triangle?

That is, in the figure, `a` and `theta` are given. Need to compute `b` and `c`.

It is noted that given one angle, the ratio of any two sides of similar triangle is a constant.

In the figure

 •  `a/c = m/o = p/r =` constant

 •  `b/c = n/o = q/r =` a different constant

 •  `a/b = m/n = p/q =` another constant

trigonometric ratios of similar triangle When specifying the ratio of the sides, two sides out of the three sides of the triangles are used. It can either be `a/c`, or `b/c` or `a/b`.

To identify which sides are used in a given ratio, the sides are named with respect to the right angle and to the given angle.

The sides are called

 •  hypotenuse in reference to the right-angle,

 •  opposite side in reference to the given angle, and

 •  adjacent side in reference to the given angle.

Given set of right angled triangles defined by an angle `theta`.

 •  position of right angle is chosen.

 •  the position of the given angle is chosentrigonometric hypotenuse, opposite side, adjacent side  •  the side opposite to the right angle is fixed and called "hypotenuse"

 •  the side opposite to the given angle is fixed and called "opposite side"

 •  the side adjacent to the given angle is fixed and called "adjacent side"

If the given angle is chosen differently...trigonometric hypotenuse, opposite side, adjacent side  •  the side opposite to the right angle is "hypotenuse"

 •  the side opposite to the given angle is "opposite side"

 •  the side adjacent to the given angle is "adjacent side"

In a right angled triangle, When an angle and one of the sides is specified, the ratio of length of sides is constant.

 •  Ratio between opposite side and hypotenuse is required if hypotenuse is given and opposite side is to be computed (or vise versa).

 •  Ratio between adjacent side and hypotenuse is required if hypotenuse is given and adjacent side is to be computed (or vise versa).

 •  Ratio between opposite side and adjacent side is required if opposite side is given and adjacent side is to be computed (or vice versa).

These three ratios named as `sin, cos, tan`.

For a set of similar right angled triangle specified by angle `theta`, the following ratios are defined. These are some of the trigonometric ratiostrigonometric hypotenuse, opposite side, adjacent side Trigonometric Ratios:

`sin theta = text(opposite)/text(hypotenuse)`

`cos theta = text(adjacent)/text(hypotenuse)`

`tan theta = text(opposite)/text(adjacent)`

In the next set of pages, why the ratios are named as `sin`, `cos`, and `tan` is explained. Pl. continue...

The word trigonometry, `color(coral)(text(trigon)) + color(deepskyblue)(text(metry)),``text( means ) color(coral)(text(triangle )) + color(deepskyblue)(text( measurement procedures))`

Trigonometric Ratios for an angle are defined as ratio of sides of Right Angled Triangle.

Trigonometric Ratios:

`sin theta = text(opposite)/text(hypotenuse)`

`cos theta = text(adjacent)/text(hypotenuse)`

`tan theta = text(opposite)/text(adjacent)`

familiarize with the terminology
sine

familiarize with the terminology
cos

familiarize with the terminology
tan

Note that when `sin A` is written, it does not mean four variables `s`, `i`, `n` and `A`. The same can be explained for `cos A` and `tan A`
`sin A` is a function of variable `A`.

                            
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