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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.

A circle with radius `1` unit is called an unit circle.

right angled triangle in unit circle 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`.

The hypotenuse in the given triangle is `bar(OQ)`.

The `bar(OP)` is the adjacent side and `bar(PQ)` is the opposite side to `/_theta`

chord in a unit circle The chord to the circle is `bar((Q Q′)`.

chord in a unit circle Note that in the given unit circle `bar(OQ)= 1`.

`sin theta` in the given figure is `bar(PQ) -: bar(OQ)` ` = bar(PQ)`

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

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

`sin theta = 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`.

The complementary angle of an angle `theta` is `90-theta`.

complementary angle in unit circle The complementary angle for `/_POQ = theta` 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'.

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

`cos theta = x`

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

tangent to unit circle Consider two triangles `OPQ` and `OTS`. It is noted that `bar(TS)` is a tangent to the circle.

tangent to unit circle The triangles `/_\ OPQ` and `/_\ OTS` are similar right-angled-triangles. And `bar(OT)=bar(OQ)=1` as it is unit circle.

ratios of sides of similar triangles are equal
`bar(ST) -: bar(OT) = bar(PQ) -: bar(OP)`

substituting `bar(OT)=1`, `bar(PQ) = sin theta` and `bar(OP)=cos theta`
`bar(ST) = (sin theta)/(cos theta)`
`bar(ST) = tan theta`

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)`

secant in a unit circle Consider the line `bar(S′S)`. It is a secant to the circle.

trigonometric secant in unit circle Consider the circle with part of the secant `OS`

`/_\ 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`

the line segment
`bar(OS)`
`= 1/(cos theta)`
`= sec theta`

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`

The line segment `bar(OT)`

`= 1/(sin theta)`

`= text(cosec) theta text( or ) csc theta`

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`.

The `bar(QT)`

`=(cos theta)/(sin theta)`

`=cot theta`

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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