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Thought-Process to Discover Knowledge

Welcome to nubtrek.

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.

Read in the blogs more about the unique learning experience at nubtrek.continue
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`.

What is the value of `sin 1^@`? Can you memorize trigonometric ratios for all angles?

  • `0.0175`
  • `0.0157`
  • one cannot memorize all values of `sin theta`.
  • one cannot memorize all values of `sin theta`.

The answer is 'one cannot memorize all values of `sin theta`'. A reference table is to be created and used when required.

For which angles can 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.
  • for the angles for which the ratio between two sides of triangles can be computed using only given angle.
  • For none

The answer is 'for the angles for which the ratio between two sides of triangles can be computed using only given angle'

In which of the following, can the ratio between two sides of triangles be computed?

  • equilateral triangles
  • isosceles triangles
  • triangles with one angle `0^@`
  • all the above
  • all the above

The answer is 'all the above'

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. trigonometry standard angles from equilateral triangles What standard angles can be derived from equilateral triangles?

  • `60^@`
  • `30^@`
  • both the above
  • both the above

The answer is 'both the above'

The two sides of the isosceles right angled triangle are same and the hypotenuse is `1`.trigonometry standard angles from isosceles right angled triangles What standard angles can be derived from isosceles right-angled-triangles?

  • `0^@`
  • `45^@`
  • `45^@`
  • `90^@`

The answer is '`45^@`'

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`. trigonometry standard angles from triangle with 0 degree What standard angles can be derived from a right-angled triangle with one of the angles `0^@`?

  • `0^@`
  • `90^@`
  • both the above
  • both the above

The answer is 'both the above'

What does 'standard' mean?

  • common and established as norm
  • special and distinctively specified

Answer is '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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