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Thought-Process to Discover Knowledge
Welcome to the only place where the essence of trigonometry is explained.
• a right-triangle is specified by 2 independent parameters.
• That means, if an angle and the length of a side is given, then one should be able to calculate the length of the other two sides.
• What property one can use to calculate the above? For a given angle, the ratio of sides is a constant.
Thus, the ratio of sides comes into existence as `sin`, `cos`, `tan` etc. The thought-process is revolutionary and aweinspiring.
Beyond the definitions of trigonometric ratios, the following are covered.
• trigonometric ratios for standard angles
• trigonometric identities based on Pythagoras Theorem
(click for the list of lessons in this topic)
Basics of Trigonometry
In this lesson, first the basics required to understand trigonometry are revised. Then, the revolutionary and aweinspiring explanation of trigonometric ratios is provided.
• A triangle has `6` or `7` parameters (`3` sides, `3` angles, sometimes height as the `7`th parameter)
• It has `3` independent parameters, meaning the other parameters can be calculated from the given `3`.
• A right triangle has `2` independent parameters, as one angle is already given as `90^@`.
• Given an angle and a side (which are the `2` independent parameters), how to compute the other two sides?
• The answer is that, given an angle, the ratio of two sides is a known constant. So, the known constant and the given side are used to compute the other sides.
This leads to defining the trigonometric ratios (ratio of sides as known constant) `sin`, `cos`, `tan`.
Trigonometric Ratios for Standard Angles
Learn something made astoundingly easy: Trigonometric ratios for standard angles, derived from
• equilateral triangles `30^@` and `60^@`,
• isosceles right-triangles `45^@`, and
• a triangle with one-side zero `90^@` and `0^@`.
Once the basis of standard angles is understood as the different triangles given above, the values of trigonometric ratios are very very easy to quickly calculate. No need to blindly memorize.
Trigonometric Identities and Complementary Angles
In this topic, some basic patterns and relations are explained in very simple thought process. The Pythagorean trigonometric identities and trigonometric values for complementary angles are explained.