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Circular measure. Definition of p Definition of radians. Unit 4:Mathematics. Aims Introduce radians and circular theorem. Objectives Identify parts of a circle and calculate triangles within a circle. Calculate circular and segment measures. Re-Call.

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Circular measure

Circular measure

Definition of p

Definition of radians


Unit 4 mathematics
Unit 4:Mathematics

Aims

  • Introduce radians and circular theorem.

Objectives

  • Identify parts of a circle and calculate triangles within a circle.

  • Calculate circular and segment measures.


Re call
Re-Call

  • sine x = (side opposite x)/hypotenuse cosine x = (side adjacent x)/hypotenuse

    tangent x =(side opposite x)/(side adjacent x)

  • sin A = a/c, cosine A = b/c, & tangent A = a/b.


Re call1
Re-Call

  • The reciprocal ratios are trigonometric ratios, too.  They are outlined below.

  • cotangent x = 1/tan x = (adjacent side)/(opposite side)

  • secant x = 1/cos x = (hypotenuse)/(adjacent side)

  • cosecant x = 1/sin x = (hypotenuse)/(opposite side)


Definition of
Definition of

  • Take any size of circle.

Cis the Circumference,

the distance around the outside.

c

d

d is the Diameter.


  • Cis the Circumference, the distance around the outside.

  • d is the Diameter

C

1

2

3

d

C ynymwneud3gwaithd

Cis about 3 times d


C

1

2

3

d

C = 3.1415927 x d

=3.1415927 (pi)

This is true for any size of circle


Definition of radians
Definition of radians

s

 =

r

+

s

r




Calculate the measure of the arc length s in the circle pictured below
Calculate the measure of the arc length below?s in the circle pictured below?


Definition below?

radians

S

 =

r

 = 2p radians

r

S = 2pr


Radians degrees
radians below?_Degrees


  • Circumference below?

  • The circumference of a circle is the perimeter of the circle

GylcheddCircumference


  • The below?diameter of a circle is a line across the circle which passes through the centre.

  • Radius

    • The radius of a circle is the distance from the centre of the circle to any point on the circumference. The radius is half the length of the diameter


  • Segment below?

    • A chord divides a circle into two segments: a minor segment and a major segment.

Chord

minor

Segment

bach

Tant chord

major

Segment

mawr


Tangent below?

  • A tangent is a line which touches the circumference of a circle at one point only and is parallel to the circumference at that point.

  • An arc is part of the circumference of a circle.

  • A sector is formed between 2 radii and the circumference

arc

tangent

sector


Properties of a circle
Properties below?of a Circle

  • The angle in a semi-circle is always a right angle

    • If 2 chords are drawn from a point on the circumference of a circle to each end of a diameter the angle between the two chords is always a right angle.


  • The below?angle at the centre of a circle

    = twice the angle at the circumference

If lines are drawn from each end of a chord to a point on the circumference of a circle and to the centre, the angle at the centre is twice the angle at the circumference.


  • Angles below?in the same segment are equal

If two chords are drawn from a point on the circumference of a circle to each end of a third chord the intersecting angle is the same no matter where the point is providing the points are in the same segment of the circle.


Opposite angles in a cyclic quadrilateral are supplementary
Opposite below?angles in a cyclic quadrilateral are supplementary

A cyclic quadrilateral is a quadrilateral whose vertices lie on the circumference of a circle.

The opposite angles of a cyclic quadrilateral add up to 180


Tangents
Tangents below?

  • A radius drawn from the point where a tangent touches a circle is perpendicular (at 90) to the tangent.


  • Tangents below?drawn to a circle from the same point outside the circle are equal in length. PA = PB

  • OP bisects angle APB

A

P

O

B


  • B below?is the centre of the circle. Find angles BPC, BCP, ABP and PAB.

  • BP = BC,  BPC = BCP

  • (isosceles triangle)

    • BPC + BCP = 180 – 66 = 114

    • BPC = BCP = 57

    • ABP = 180 – 66 = 114

    • (angles on a straight line = 180)

    • PAB = ½ × 66 = 33

    • (angle at centre of a circle = twice that at the circumference)

  • P

    66

    A

    C

    B


    • Calculate below?angles p, q and r.

      p + 85 = 180 (opposite angles in a cyclic quadrilateral)

    • p = 180 – 85 = 95

    • q + 101 = 180 (opposite angles in a cyclic quadrilateral)

    • q = 180 – 101 = 79

    • r + q = 180 (angles on a

    • straight line)

    • r = 180 – 79 = 101

    r

    q

    p

    85

    101


    • 3. below?Work out angles a, b, c and d.

    • a + 41 + 101 = 180 (angles in a triangle)

    • a = 180 – 41 – 101 = 38

    • b = 101 (opposite angles)

    • d = 41 (angles in the same segment)

  • c = 38

  • (angles in thesamesegment)

  • a

    41

    101

    b

    c

    d


    • 4. below?Work out angles a, b and c.

    • c = 90 (angles in a semicircle)

    • a + 59 + 90 = 180 (angles in a triangle)

    • a = 180 – 59 – 90 = 31

    • a + b = 90 (radius is

    • perpendicular to the tangent)

    • b = 90 – 31 = 59

    c

    59

    b

    a


    P below?

    • 5. Calculate angles XPY and OXY .

    • PX = PY  PYX =  PXY

    • (isosceles triangle)

    • PYX = 75

    • PXO = 90 (radius is

    • perpendicular to the tangent)

    • OXY = 90 – 75 = 15

    • XPY = 180 – 75 – 75 = 30

    • (angles in a triangle)

    75

    X

    Y

    O


    • 6. below?XTY is a tangent to the circle, centre O. P and Q are points on the circumference. OQ is parallel to PT. Angle QOT = 37. Find angles OPT and PTY.

    • OTP = 37 (alternate angles)

    • OP = OT  OPT = OTP = 37(isosceles triangle)

    • OTY = 90 (radius is

    • perpendicular to the tangent)

    • PTY = OTY - OTP

    • PTY = 90 – 37 = 53

    O

    37

    Q

    P

    T

    Y

    X


    7. below?PTR is a tangent to the circle, centre O. The chord AB is parallel to PR. X is a point on the circumference. Angle ORT = 18.

    • Work out angle AXB.

    • OTR = 90 (radius is perpendicular to the tangent)

    • TOR = 180 – 90 – 18 = 72 (angles in a triangle)

    • TAB = ½ × 72 = 36 (angle at circumference = ½ angle at centre).

    • ATP = 36 (alternate angles)

    • ATO = 90 – 36 = 54

    X

    O

    B

    A

    18

    R

    T

    P


    • below?OTR = 90, TOR = 72, TAB = 36,

    • ATP = 36, ATO = 54

  • OT = OB  OTB = OBT

    • (isosceles triangle)

    • OTB + OBT = 180 - TOR = 180 – 72 = 108

    • OTB = ½ × 108 = 54

    • ATB = ATO + OTB

    • ATB = 54 + 54 = 108

    • AXB = 180 – 108 = 72

    • (angles in a cyclic

    • quadrilateral)

  • X

    O

    B

    A

    18

    R

    T

    P


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