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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.

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. 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?

• bar((Q Q′)
• bar((Q Q′)
• bar(OP)

The answer is 'bar((Q Q′)'

What is sin theta in the given figure? 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.

Given that the point Q is (x,y), what is sin theta?

• 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. 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?

• /_POR
• /_QOR
• /_QOR

The answer 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), what is cos theta?

• 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.

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

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

• bar(QS)
• bar(S′S)
• bar(S′S)

The answer is 'bar(S′S)'

What is the length of bar(OS)? 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. 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. 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. 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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