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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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mathsTrigonometryBasics of Trigonometry

Basics of Triangles

In this page, the basics of triangles required to understand trigonometry is revised.



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What is the sum of interior angles of a triangle?

  • `90^@`
  • `180^@`
  • `180^@`
  • `270^@`
  • `360^@`

The answer is '`180^@`'

What are "Congruent Triangles"?congruent triangles

  • Triangles that are identical
  • Triangles that exactly fit over
  • both the above
  • both the above

The answer is 'both the above'. Note triangles can be flipped or rotated while comparing.

What are "Similar Triangles"?similar triangles

  • Triangles that have same shape
  • Triangles that are scaled up or down version of one another
  • both the above
  • both the above

The answer is 'both the above'

What is a right angled triangle?right angled triangle

  • A triangle with one angle `90^@`
  • A triangle with one angle `90^@`
  • A triangle with all angles equal.
  • both the above

The answer is 'A triangle with one angle `90^@`'

How many independent parameters define a triangle?

  • `1`
  • `2`
  • `3`
  • `3`
  • `4`

The answer is '`3`'.

For example, given that the sides of a triangle are `1`, `1`, and the angle between the sides is `60^@`, it is derived that the triangle is a equilateral triangle.

The independent parameters are given

 •  side-side-side

 •  side-angle-side

 •  angle-side-angle

 •  angle-angle-side

Note that, angle-angle-angle are not independent parameters as sum of angles in a triangle is `180^@`. So given two angles, third angle can be calculated.

How many independent parameters define a right angled triangle?

  • `1`
  • `2`
  • `2`
  • `3`
  • `4`

The answer is '`2`'.

For a right angled triangle, one angle is given as `90^@`, so the independent parameters are given

 •  side-side

 •  side-angle

What does "two independent parameters define a right angled triangle" mean?

  • If two of { 2 angles (excluding the right angle) and 3 sides} are given, then all other angle or sides are exactly defined and can be mathematically computed.
  • If two of { 2 angles (excluding the right angle) and 3 sides} are given, then all other angle or sides are exactly defined and can be mathematically computed.
  • Two independent parameters define set of similar triangles.

The answer is "If two of `2` angles (excluding the right angle) or `3` sides are given, then all other angle or sides are exactly defined and can be mathematically computed".

The combination of two parameters are

Given side-side, the third side and two angles can be calculated.

Given angle-side, the other two sides and the angle can be calculated.

The two arms to the right angle are given as `a` and `b`. Since two independent parameters define a right angled triangle, what is the third side `c`?

  • `c = sqrt(a^2+b^2)`
  • `c = sqrt(a^2+b^2)`
  • can not be computed

The answer is "`c = sqrt(a^2+b^2)`". This is the Pythagoras theorem.

Pythagoras Theorem: It is understood that the two arms of right angled triangle defines the hypotenuse. As length of one arm changes, the hypotenuse also changes accordingly. The relationship between the arms and the hypotenuse is given by `c^2 = a^2+b^2`.

This happens to be a cute expression and it is used often in mathematics. Thus, it is one of the most popular theorems. The explanations like `c^2` represents area of square with hypotenuse as side etc. are proofs or analogies of this relationship.

The expression, in absolute sense, captures the relationship between two arms and the hypotenuse, as only two parameters are independent.

Pythagoras Theorem (defined in first principles): A right angled triangle is defined by two independent parameters. Hence, The three parameters, hypotenuse and the two legs, are related. The relationship is given by the expression
`c^2 = a^2+b^2`

where `c` is the hypotenuse and `a` and `b` are the two legs.

How many independent parameters define the set of similar right angled triangles?set of similar right angled triangles (Figure shows three triangles, solid line, dotted line, and shaded.)

  • `1`
  • `1`
  • `2`
  • `3`
  • `4`

The answer is '`1`'.

Set of similar right angled triangles is defined by one angle.

Basics of Triangles to learn trigonometry :

 •  Congruent and Similar Triangles.

 •  three parameters define a triangle

 •  two parameters define a right angled triangle

 •  one parameter define set of similar right angled triangles : angle

Congruent Triangles: Triangles that are identical.

Similar Triangles: Triangles that are scaled and have same shape.

Right Angled Triangle: Triangles having one angle as `90^@`.

Number of Parameters Defining a Triangle: Minimum three independent parameters are required to specify a triangle: SSS, SAS, ASA, AAS

Number of Parameters Defining a Right Angled Triangle: Minimum two independent parameters are required to specify a right angled triangle : SA or SS

Number of Parameters Defining set of Similar Right Angled Triangles: One parameter define set of similar right angled triangles : A

                            
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