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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.

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

In this page, the trigonometric values in unit circle form is introduced and explained.



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In the previous lessons, the definitions of trigonometric ratios were explained for right angled triangles. Why these were named as

 •  sine,

 •  cosine,

 •  tan (tangent),

 •  secant,

 •  co-secant, and

 •  cot (co-tangent)

It is important to learn this part to properly connect and retain the knowledge.

What is an unit circle?

  • A circle with radius=`1`
  • A circle with radius=`1`
  • A circle with area =`1`
  • A circle with perimeter = `1`

The answer is 'A circle with radius=`1`'

It was explained that trigonometric ratios are defined for set of similar right angled triangles. Consider the right angled triangle made within a unit circle as given in the figure. Any right angled triangle with one angle `theta` is represented by the `/_\ OPQ`. right angled triangle in unit circle What is the hypotenuse in the given triangle?

  • `bar(OQ)`
  • `bar(OQ)`
  • `bar(OP)`
  • `bar(PQ)`

The answer is '`bar(OQ)`'.

Similarly, the `bar(OP)` is the adjacent side and `bar(PQ)` is the opposite side to `/_theta`.

Which of the following is a 'chord' to the circle?chord in a unit circle

  • `bar((Q Q′)`
  • `bar((Q Q′)`
  • `bar(OP)`

The answer is '`bar((Q Q′)`'

What is `sin theta` in the given figure?chord in a unit circle Note that in the given unit circle `bar(OQ)= 1`.

  • `bar(PQ) -: bar(OQ)`
  • `bar(PQ)`
  • Both the above
  • Both the above

The answer is 'Both the above'.

The root word of `sin` refers to chord of a circle.

For a given angle `theta`, the line `bar(PQ)` is half of the chord as shown in figure. So the ratio is named as `sin` referring the relation to length of the chord at the given angle.sine in a unit circle

Given that the point `Q` is `(x,y)`, what is `sin theta`?projection form unit circle sin

  • `x`
  • `y`
  • `y`
  • Not possible to find

The answer is '`y`'.

So far, `sin, cos, ...` were referred as trigonometric ratios. Considering the unit circle and the point `(x, y)` at angle `theta` on the unit circle, `sin, cos ...` are referred as trigonometric values.projection form unit circle sin `sin theta` is the projection on y-axis for the line of unit length at angle `theta`.

What is the complementary angle of `theta`?

  • `90-theta`
  • `90-theta`
  • `180-theta`

The answer is '`90-theta`'

Which of the following is the complementary angle for `theta`?complementary angle in unit circle

  • `/_POR`
  • `/_QOR`
  • `/_QOR`

The answer is '`/_QOR`'

For a given angle `theta`, the `sin` of complementary angle is `cos` or cosine.cosine in unit circle co-sine is the short form of 'complementary sine'.

Given that the point `Q` is `(x,y)`, what is `cos theta`?projection form unit circle cos

  • `x`
  • `x`
  • `y`
  • Not possible to find

The answer is '`x`'.

`cos theta` is the projection on x-axis for the line of unit length at angle `theta`.projection form unit circle cos

Consider two triangles `O P Q` and `O T S`. Which of the following is a tangent to the circle?tangent to unit circle

  • `bar(OP)`
  • `bar(QR)`
  • `bar(TS)`
  • `bar(TS)`

The answer is '`bar(TS)`'

Which of the following is the length of line segment `bar(TS)`?tangent to unit circle Note that the triangles `/_\ OPQ` and `/_\ OTS` are similar right-angled-triangles. And `bar(OT)=bar(OQ)=1`.

  • `bar(ST) = bar(PQ) -: bar(OP)`
  • `bar(ST) = (sin theta)/(cos theta)`
  • both the above
  • both the above

The answer is 'both the above'

For a given angle `theta`, the `tan theta` is the length of the line segment on tangent as shown in figure.tan in unit circle `tan` is the short form of 'tangent'.

The `tan` of an angle is equivalently given as `bar(QS′)` as shown in figure. tan in unit circle Note the following:
`/_\ OPQ` and `/_\ OQS′` are similar triangles and so
`bar(QS′)/bar(OQ) = bar(PQ)/bar(OP)`
so, `tan theta = bar(PQ)/bar(OP)`

Which of the following is a "secant" to the circle?secant in a unit circle

  • `bar(QS)`
  • `bar(S′S)`
  • `bar(S′S)`

The answer is '`bar(S′S)`'

What is the length of `bar(OS)`?trigonometric secant in unit circle Note that `/_\ OPQ` and `/_\ OQS` are similar right angled triangles.
So `text (hypotenuse) -: text(adjacent)` for the triangles are equal.

`bar(OS) -: bar(OQ) = bar(OQ) -: bar(OP)`

Substitute `bar(OQ) = 1`, and `bar(OP) = cos theta`

  • `1/(cos theta)`
  • `1 / x`
  • `1/bar(OP)`
  • all the above
  • all the above

The answer is 'all the above'

The trigonometric value corresponding to `bar(OS)` is `sec theta`.trigonometric secant in unit circle `sec theta = 1/(cos theta) = 1/x`

The secant for the complementary angle of theta is `bar(OT)` as given in the figure. It is noted that `/_\ TQO` and `/_\ OPQ` are similar right-angled-triangles.

ratio of corresponding sides are equal `bar(OT)/bar(OQ) = bar(OQ)/bar(PQ)`

substitute `bar(OQ) = 1` and `bar(PQ) = sin theta`. What is the length of `bar(OT)`?

  • `1/bar(PQ)`
  • `1/(sin theta)`
  • `1/y`
  • all the above.
  • all the above.

The answer is 'All the above'

The trigonometric value corresponding to `bar(OT)` is `text(cosec) theta`.trigonometric co-secant in unit circle `text(cosec) theta = 1/(sin theta) = 1/y`

The tangent for the complementary angle of theta is `bar(QT)` as given in the figure. It is noted that `/_\ TQO` and `/_\ OPQ` are similar right-angled-triangles.

ratio of corresponding sides are equal `bar(QT)/bar(OQ) = bar(OP)/bar(PQ)`

substitute `bar(OQ) = 1`, `bar(PQ) = sin theta` and `bar(OP) = cos theta`. What is the length of `bar(QT)`?

  • `bar(OP)/bar(PQ)`
  • `(cos theta)/(sin theta)`
  • `x/y`
  • all the above.
  • all the above.

The answer is 'All the above'

The trigonometric value corresponding to `bar(QT)` is `cot theta`.trigonometric co-tangent cot in unit circle `text(cot) theta = 1/(tan theta) = (cos theta)/(sin theta) = x/y`

The trigonometric values for a given angle `theta` is defined as lengths in relation to

 •  chord or sine : `sin theta`

 •  tangent : `tan theta`

 •  secant : `sec theta`

all the above for complementary angle

 •  co-sine : `cos theta`

 •  co-tangent : `cot theta`

 •  co-secant : `csc theta` or `text(cosec) theta`

Trigonometric Values: For a line segment of unit length at the angle `theta`

summary of all trigonometric values in unit circle The point on unit circle for given angle `theta` is `P(x,y)`.

Note: `x` and `y` are the projections on `x`-axis and `y`-axis.

 •  chord or sine : `sin theta = y`

 •  tangent : `tan theta = y/x`

 •  secant : `sec theta = 1/x`

For complementary-angle

 •  co-sine : `cos theta = x`

 •  co-tangent : `cot theta = x/y`

 •  co-secant : `csc theta` or `text(cosec) theta = 1/y`

                            
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