nubtrek

Server Error

Server Not Reachable.

This may be due to your internet connection or the nubtrek server is offline.

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 Identities and Complementary Angles

Trigonometric Ratios for Complementary Angles

The relationship between trigonometric ratios of an angle and its complementary angle is explained.



click on the content to continue..

What is the complementary angle of `theta`?

  • angle that completes `theta` to a right angle
  • `90-theta`
  • both the above
  • both the above

The answer is 'both the above'

In the right angled triangle shown in figure, if `/_P = theta` then what is `/_R` ? angles ratio

  • complementary angle of `theta`
  • `90-theta`
  • both the above
  • both the above

The answer is 'both the above'

In the right angled triangle shown in figure, what is the opposite side for `/_P ` ? angles ratio

  • `PQ`
  • `QR`
  • `QR`

The answer is '`QR`'

In the right angled triangle shown in figure, what is the adjacent side for `/_R ` ? angles ratio

  • `PQ`
  • `QR`
  • `QR`

The answer is 'QR'

A right angled triangle is shown in figure.angles ratio Opposite of `/_theta` = Adjacent of `/_(90-theta)`
Adjacent of `/_theta` = Opposite of `/_(90-theta)`

The trigonometric ratios of `90-theta` can be expressed in terms of trigonometric ratios of `theta` or vice versa. angles ratio complementary angles From the figure
`sin theta = (RQ)/(PR)`
`cos theta = (PQ)/(PR)`
`sin (90-theta) = (PQ)/(PR)`
`cos (90-theta) = (RQ)/(PR)`

Comparing the above, it is concluded that
`sin (90-theta) = cos theta`
`cos (90-theta) = sin theta`

It is noted that the two angles in a right angles triangle are complementary angles and the trigonometric ratios of one angle are related to the trigonometric ratios of the other angle.

That is

 •  hypotenuse is common for both the angles

 •  opposite side for an angle is the adjacent side for the other angle

 •  adjacent side for an angle is the opposite side for the other angle

This relation is used to derived Trigonometric Ratios for Complementary Angles .

Trigonometric Ratios for Complementary Angles:
`sin(90-theta) = cos theta`
`cos(90-theta) = sin theta`

The trigonometric ratios of `90-theta` can be expressed in terms of trigonometric ratios of `theta` or vice versa. angles ratio complementary angles From the figure
`tan theta = (RQ)/(PQ)`
`cot theta = (PQ)/(RQ)`
`tan (90-theta) = (PQ)/(RQ)`
`cot (90-theta) = (RQ)/(PQ)`

Comparing the above, it is concluded that
`tan (90-theta) = cot theta`
`cot (90-theta) = tan theta`

`tan(90-theta) = cot theta`
`cot(90-theta) = tan theta`

The trigonometric ratios of `90-theta` can be expressed in terms of trigonometric ratios of `theta` or vice versa. angles ratio complementary angles From the figure
`csc theta = (PR)/(RQ)`
`sec theta = (PR)/(PQ)`
`csc (90-theta) = (PR)/(PQ)`
`sec (90-theta) = (PR)/(RQ)`

Comparing the above, it is concluded that
`csc(90-theta) = sec theta`
`sec(90-theta) = csc theta`

`csc(90-theta) = sec theta`
`sec(90-theta) = csc theta`

Solved Exercise Problem:

Which of the following equals `sin 36`?

  • `cos 54`
  • `cos 54`
  • `tan 54`
  • `sec 54`
  • `sin 54`

The answer is '`cos 54`'

Solved Exercise Problem:

`(tan 75) / (cot 15) = ?`

  • `tan^2 75`
  • `1`
  • `1`

The answer is '`1`' As `cot 15` `=tan(90-15)` `=tan 75`.

Solved Exercise Problem:

Which of the following equals `sin 36 - cos 36 + sin 54`?

  • `sin 54`
  • `sin 36`
  • `sin 36`

The answer is '`sin 36`'.

                            
switch to slide-show version