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Thought-Process to Discover Knowledge

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mathsAdvanced TrigonometryTrigonometric 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.

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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.

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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