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Thought-Process to Discover Knowledge

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Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

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mathsTrigonometryTrigonometric Ratios for Standard Angles

Understanding Standard Angles

What are the standard angles for which trigonometric ratios are defined? These angles are chosen because of some pattern or properties. This page explains the reason why some angles are special.

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We had studied that the ratio of two sides of a right-triangle is a constant and the such constants are named `sin theta, cos theta, cdots` for given values of `theta = 0^@, 0.2^@, 1^@, 2.7^@, cdots`.

It is noted that one cannot memorize all values of `sin theta`. A reference table is to be created and used when required.

Students work out values of trigonometric ratios `sin`, `cos`, and `tan`, for the angles for which the ratio between two sides of triangles can be computed using only given angle

The ratio between two sides of triangles be computed for the following

 •  equilateral triangles

 •  isosceles triangles

 •  triangles with one angle `0^@`

trigonometry standard angles from equilateral triangles The equilateral triangle is split into two right angled triangles as shown in the figure. The hypotenuse `bar(OP) = 1`. Then the side `bar(OQ) = 1/2`. The other side is to be computed using Pythagoras Theorem.

The standard angles in this triangles are `60^@` and `30^@`

trigonometry standard angles from isosceles right angled triangles The two sides of the isosceles right angled triangle are same and the hypotenuse is `1`.

The standard angle derived from isosceles right-angled-triangles is `45^@`.

trigonometry standard angles from triangle with 0 degree The figure shows a triangle `Delta OPQ` with `/_P = 0^@`. Imagine the `bar(PR)` moves towards `bar(PO)` and forms the triangle where point `R` meets point `O`.

The standard angles in a right-angled triangle with one of the angles `0^@` are `0^@` and `90^@`.

The word "standard" means: common and established as norm.

The standard angles for which the trigonometric ratios are in the form of simple ratios, are derived from equilateral, isosceles, and `0^@` triangles. Use the known properties of these triangles to compute the trigonometric values for the standard angles.

Trigonometric Ratios for Standard Angles:

 •  `0^@` and `90^@` : right-angled-triangle with two sides `1` and the third side `0`.

 •  `45^@` : isosceles right-angled triangle.

 •  `30^@` and `60^@` : half of equilateral triangle making a right-angled-triangle.

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