__maths__>__Advanced Trigonometry__>__Trigonometric Values for all angles in unit circle form__### Trigonometric Values: Unit Circle Form

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

*click on the content to continue..*

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.

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`

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

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.

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. `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`.

The complementary angle for `/_POQ = theta` is `/_QOR`.

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

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`.

Consider two triangles `OPQ` and `OTS`. It is noted that `bar(TS)` is a tangent to the 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` is the short form of 'tangent'.

The `tan` of an angle is equivalently given as `bar(QS′)` as shown in figure. 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)`

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

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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