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summary of this topic

### Trigonometric Values for all angles in unit circle form

Voice

Voice

Home

Names of Trigonometric Functions

»  Angle theta specifies a class of similar right-triangles.
→  Any one representative triangle captures the properties of all in the class.

»  The representative triangle is chosen in the unit circle.

»  sine: root word meaning chord.
»  tangent: meaning touching the curve
»  secant: meaning cutting through

»  complementary: meaning completes to right angle.
»  co-sine
»  co-tangent
»  co-secant

Trigonometric Values Redefined

»  Point on unit circle at the given angle theta
Representative of the class of similar right-triangles

»  Projection on y-axis = y
→  opposite side in triangle is generalized to the y-coordinate

»  Projection on x-axis = x
→  adjacent side in triangle is generalized to the x-coordinate

»  sin theta = y
y-coordinate of the point on unit circle

»  cos theta = x
x-coordinate of the point on unit circle

»  and accordingly tan, sec, csc, cot.

### Trigonometric Values: Unit Circle Form

plain and simple summary

nub

plain and simple summary

nub

dummy

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

simple steps to build the foundation

trek

simple steps to build the foundation

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

Keep tapping on the content to continue learning.
Starting on learning "Trigonometric Values: Unit Circle Form". ;; In this page, the trigonometric values in unit circle form is introduced and explained.

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 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 Delta OPQ. What is the hypotenuse in the given triangle?

• 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(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

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
• 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
• 180-theta

The answer is '90-theta'

Which of the following is the complementary angle for theta?

• /_POR
• /_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
• 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.

Which of the following is a tangent to the circle?

• bar(OP)
• bar(QR)
• bar(TS)

The answer is 'bar(TS)'

Which of the following is length of line segment bar(TS)? Note that the triangles Delta OPQ and Delta 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

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:
Delta OPQ and Delta 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)

The answer is 'bar(S′S)'

What is the length of bar(OS)? Note that Delta OPQ and Delta 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

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. What is the length of bar(OT)? It is noted that Delta TQO and Delta 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.

• 1/bar(PQ)
• 1/(sin theta)
• 1/y
• 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. What is the length of bar(QT)? It is noted that Delta TQO and Delta 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.

• bar(OP)/bar(PQ)
• (cos theta)/(sin theta)
• x/y
• 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

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

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

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

Progress

Progress

In the previous lessons, the definitions of trigonometric ratios were explained for right angled triangles. Why these were named as sine, co-sine, tangent, secant, co-secant, and co-tangent? It is important to learn this part to properly connect and retain the knowledge.
What is an unit circle?
area
A circle with area one
perimeter
A circle with perimeter one
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 triangle O P Q. ;; What is the hypotenuse in the given triangle?
OQ
line O Q
OP
line O P
PQ
line P Q
The answer is 'line OQ'. Similarly, the line O P is the adjacent side and line P Q is the opposite side to the angle theta
Which of the following is a 'chord' to the circle?
Q Q;QQ; Q ;queue; prime
line Q Q prime
O P
line O P
The answer is 'line Q Q prime'
What is sine theta in the given figure? Note that in the given unit circle line O Q equals 1.
OQ
line PQ by line O Q
PQ
line P Q
both;above
Both the above
The answer is 'Both the above'.
The root word of sine refers to chord of a circle. For a given angle theta, the line PQ is half of the chord as shown in figure. So the ratio is named as sine referring the relation to length of the chord at the given angle.
Given that the point Q is x y, what is sine theta?
x;ex
x
y;why
y
not;possible;find
Not possible to find
So far, sine, cause, et cetera were referred as trigonometric ratios. Considering the unit circle and the point x, y at angle theta on the unit circle, sine, cause, et cetera are referred as trigonometric values. ;; sine theta is the projection on y-axis for the line of unit length at angle theta .
What is the complementary angle of theta ?
90;ninety
90 minus theta
180;eighty
180 minus theta
The answer is '90 minus theta'
Which of the following is the complementary angle for theta ?
P; pee
angle P O R
Q; queue
angle Q O R
The answer is 'angle Q O R'
For a given angle theta, the sine of complementary angle is cause or cosine. co-sine is the short form of 'complementary sine'.
Given that the point Q is x y, what is cause theta?
x;ex
x
y;why
y
not;possible;find
Not possible to find
cos theta is the projection on x-axis for the line of unit length at angle theta.
Which of the following is a tangent to the circle?
o;p;pee
line O P
Q;queue;are;r
line Q R
t;s;yes;tee;tea
line T S
The answer is 'line T S'
Which of the following is length of line segment T S ? Note that the triangles O P Q and O T S are similar right-angled-triangles. And line O T equals line O Q equals 1.
P;Q;pee;queue;oh;O
line S T = line P Q by line O P
sine;cause;cos;theta
line S T = sine theta by cause theta
both;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 line QS prime as shown in figure. Note the following: ;; triangle OPQ and triangle OQS prime are similar triangles ;; and so ;; line Q S prime by line O Q = line P Q by line O P ;; so, tan theta = line P Q by line O P
Which of the following is a "secant" to the circle?
Q; queue
line Q S
prime
line S prime S
The answer is 'line S prime S'
What is the length of line O S? ;; Note that triangle O P Q and triangle O Q S are similar right angled triangles. ;; So hypotenuse divided by adjacent side for the triangles are equal. ;; line O S divided by line O Q ;; equals ;; line O Q divided by line O P;; Substitute line O Q ;; equals ;; 1 and line O P ;; equals ;; cos theta ;;
cause; cos
1 by cos theta
x; ex
one by x
line;oh;o;p;pee
one by line O P
all;above
all the above
The answer is 'all the above'
The trigonometric value corresponding to line O S is secant theta. ;; Secant theta ;; equals ;; one by cos theta ;; equals ;; one by x.
The secant for the complementary angle of theta is line O T ;; as given in the figure. What is the length of line O T? ;; It is noted that triangle T Q O ;; and ;; triangle O P Q ;; are similar right-angled-triangles. ;; ratio of corresponding sides are equal ;; line O T by line O Q ;; equals;; line O Q by line P Q ;; substitute line O Q = 1 ;; and line P Q = sine theta. What is the length of line O T? ;;
line;pee;p;queue;q
one by line P Q
sine;theta
one by sine theta
y;why
one by y
all;above
all the above.
The answer is 'All the above'
The trigonometric value corresponding to line O T is co-secant theta. ;; co-Secant theta ;; equals ;; one by sine theta ;; equals ;; one by y.
The tangent for the complementary angle of theta is line Q T ;; as given in the figure. What is the length of line Q T? ;; It is noted that triangle T Q O ;; and ;; triangle O P Q ;; are similar right-angled-triangles. ;; ratio of corresponding sides are equal ;; line Q T by line O Q ;; equals;; line O P by line P Q ;; substitute line O Q = 1 ;; line P Q = sine theta ;; and line O P = cos theta. What is the length of line Q T? ;;
line;pee;p;queue;q;oh;o
line O P by line P Q
sine;theta;cos;cause
cos theta by sine theta
y;why;ex;x
x by y
all;above
all the above.
The answer is 'All the above'
The trigonometric value corresponding to line Q T is co-tangent theta or cot theta. ;; cot theta ;; equals ;; one by tan theta ;; equals ;; cos theta by sine theta ;; equals ;; x by y.
The trigonometric values for a given angle theta is defined as lengths in relation to ;; chord or sine ;; tangent ;; secant. ;; all the above for complementary angle ;; co-sine ;; co-tangent ;; co-secant
Trigonometric values are summarized for a unit circle.

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