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Thought-Process to Discover Knowledge

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mathsTrigonometryTrigonometric Identities and Complementary Angles

Pythagorean Trigonometric Identities

The relationship between trigonometric ratios per Pythagorean theorem is explained and referred as "Pythagorean Trigonometric Identities"



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For the triangle given in figure, What is `sin theta`?Pythagorean Trigonometric Identities illustration

  • `(OP)/(OQ)`
  • `(PQ)/(OQ)`
  • `(PQ)/(OQ)`

The answer is '`(PQ)/(OQ)`'

For the triangle given in figure, What is `cos theta`?Pythagorean Trigonometric Identities illustration

  • `(OP)/(OQ)`
  • `(OP)/(OQ)`
  • `(PQ)/(OQ)`

The answer is '`(OP)/(OQ)`'

For the triangle given in figure, which of the following represents Pythagoras theorem?Pythagorean Trigonometric Identities illustration

  • `OP^2 + PQ^2 =OQ^2`
  • `((OP)/(OQ))^2 + ((PQ)/(OQ))^2 = 1`
  • Both the above
  • Both the above

The answer is 'both the above'.

For a right angled triangle, the Pythagoras theorem is given as

`(text(opposite))^2` `+ (text(adjacent))^2` `= (text(hypotenuse))^2`

If this equation is divided by `(text(hypotenuse))^2` ; which of the following is the result?

  • `((OP)/(OQ))^2 + ((PQ)/(OQ))^2 = 1`
  • `sin^2 theta + cos^2 theta = 1`
  • Both the above
  • Both the above

The answer is 'both the above'.

Pythagorean Theorem states that in a right angled triangle, square of hypotenuse equals sum of squares of two arms. The trigonometric ratios are defined for right angled triangles. The relationships between trigonometric ratios per Pythagorean theorem are called "Pythagorean Trigonometric Identities".

`sin^2 theta + cos^2 theta = 1`

It is noted that the result is true for any value of `theta`. That is, if `theta = 27`, then `sin^2 27^@ + cos^2 27^@ = 1`

What does 'identity' mean?

  • It is just a name.
  • equality of two expressions; left and right hand side are identical
  • equality of two expressions; left and right hand side are identical

Answer is 'equality of two expressions; left and right hand side are identical'.

In the Pythagorean Trigonometric Identity `sin^2 theta + cos^2 theta = 1`, it is stated that left hand side `sin^2 theta + cos^2 theta` equals the right hand side `1`.

For a right angled triangle,

`sin^2 theta + cos^2 theta = 1`

If this equation is divided by `sin^2 theta` ; which of the following is the result?Pythagorean Trigonometric Identities illustration

  • cannot divide this equation by `sin^2 theta`
  • `1+cot^2 theta = csc^2 theta`
  • `1+cot^2 theta = csc^2 theta`

The answer is '`1+cot^2 theta = csc^2 theta`'.

`1+cot^2 theta = csc^2 theta`

For a right angled triangle,

`sin^2 theta + cos^2 theta = 1`

If this equation is divided by `cos^2 theta` ; which of the following is the result?Pythagorean Trigonometric Identities illustration

  • cannot divide this equation by `cos^2 theta`
  • `tan^2 theta + 1 = sec^2 theta`
  • `tan^2 theta + 1 = sec^2 theta`

The answer is '`tan^2 theta + 1 = sec^2 theta`'.

`tan^2 theta + 1 = sec^2 theta`

Pythagorean Trigonometric Identities:

For any theta,

`sin^2 theta + cos^2 theta = 1`

`1+cot^2 theta = csc^2 theta`

`tan^2 theta + 1 = sec^2 theta`

Note that this need not me memorized, connect these to the Pythogoras theorem and quickly derive when required.

Solved Exercise Problem:

What is the value of `1-sin^2 theta`?

  • `cos^2 theta`
  • `cos^2 theta`
  • the question is incomplete to find an answer

The answer is '`cos^2 theta`'.

This is derived from the Pythagorean trigonometric identity `sin^2 theta + cos^2 theta = 1`

Solved Exercise Problem:

What is the value of `sec^2 theta - 1`?

  • `tan^2 theta`
  • `tan^2 theta`
  • `cot^2 theta`

The answer is '`tan^2 theta`'.

This is derived from the Pythagorean trigonometric identity `tan^2 theta + 1 = sec^2 theta`

Solved Exercise Problem:

What is the value of `csc^2 theta-cot^2 theta`?

  • `cos^2 theta`
  • 1
  • 1

The answer is '`1`'.

This is derived from the Pythagorean trigonometric identity `1 + cot^2 theta = csc^2 theta`

                            
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