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summary of this topic

Trigonometric Values for all angles in unit circle form

Trigonometric Values for all angles in unit circle form

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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

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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`. right angled triangle in unit circle 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?chord in a unit circle

  • `bar((Q Q′)`
  • `bar(OP)`

The answer is '`bar((Q Q′)`'

What is `sin theta` in the given figure?chord in a unit circle 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.sine in a unit circle

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

  • `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.projection form unit circle sin `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`?complementary angle in unit circle

  • `/_POR`
  • `/_QOR`

The answer 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'.

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

  • `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`.projection form unit circle cos

Which of the following is a tangent to the circle?tangent to unit circle

  • `bar(OP)`
  • `bar(QR)`
  • `bar(TS)`

The answer is '`bar(TS)`'

Which of the following is length of line segment `bar(TS)`?tangent to unit circle 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 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:
`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?secant in a unit circle

  • `bar(QS)`
  • `bar(S′S)`

The answer is '`bar(S′S)`'

What is the length of `bar(OS)`?trigonometric secant in unit circle 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`.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. 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`.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. 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`.trigonometric co-tangent cot in unit circle `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`

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`



           

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

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Progress

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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?
radius
A circle with radius one
area
A circle with area one
perimeter
A circle with perimeter one
The answer is 'A circle with radius 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
The answer is 'y'
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
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?
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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